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10376_0 | In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of the integers. This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements. In particular, the greatest common divisor of any two elements exists and can be written as a linear combination
of them (Bézout's identity). Also every ideal in a Euclidean domain is principal, which implies a suitable generalization of the fundamental theorem of arithmetic: every Euclidean domain is a unique factorization domain.
It is important to compare the class of Euclidean domains with the larger class of principal ideal domains (PIDs). An arbitrary PID has much the same "structural properties" of a Euclidean domain (or, indeed, even of the ring of integers), but when an explicit algorithm for Euclidean division is known, one may use Euclidean algorithm and extended Euclidean algorithm to compute greatest common divisors and Bézout's identity. In particular, the existence of efficient algorithms for Euclidean division of integers and of polynomials in one variable over a field is of basic importance in computer algebra.
So, given an integral domain , it is often very useful to know that has a Euclidean function: in particular, this implies that is a PID. However, if there is no "obvious" Euclidean function, then determining whether is a PID is generally a much easier problem than determining whether it is a Euclidean domain.
Euclidean domains appear in the following chain of class inclusions:
Definition
Let be an integral domain. A Euclidean function on is a function from } to the non-negative integers satisfying the following fundamental division-with-remainder property:
(EF1) If and are in and is nonzero, then there exist and in such that and either or .
A Euclidean domain is an integral domain which can be endowed with at least one Euclidean function. It is important to note that a particular Euclidean function is not part of the structure of a Euclidean domain; in general, a Euclidean domain will admit many different Euclidean functions.
Most algebra texts require a Euclidean function to have the following additional property: | [
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10376_1 | Most algebra texts require a Euclidean function to have the following additional property:
(EF2) For all nonzero and in , .
However, one can show that (EF1) alone suffices to define a Euclidean domain. If an integral domain is endowed with a function satisfying (EF1), then can also be endowed with a function satisfying both (EF1) and (EF2) simultaneously. Indeed, for }, one can define as follows:
In words, one may define to be the minimum value attained by on the set of all non-zero elements of the principal ideal generated by .
A Euclidean function is multiplicative if and is never zero. It follows that . More generally, if and only if is a unit.
Notes on the definition
Many authors use other terms in place of "Euclidean function", such as "degree function", "valuation function", "gauge function" or "norm function". Some authors also require the domain of the Euclidean function to be the entire ring ; however this does not essentially affect the definition, since (EF1) does not involve the value of . The definition is sometimes generalized by allowing the Euclidean function to take its values in any well-ordered set; this weakening does not affect the most important implications of the Euclidean property.
The property (EF1) can be restated as follows: for any principal ideal of with nonzero generator , all nonzero classes of the quotient ring have a representative with . Since the possible values of are well-ordered, this property can be established by proving that for any with minimal value of in its class. Note that, for a Euclidean function that is so established, there need not exist an effective method to determine and in (EF1).
Examples
Examples of Euclidean domains include: | [
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] |
10376_2 | Examples
Examples of Euclidean domains include:
Any field. Define for all nonzero .
, the ring of integers. Define , the absolute value of .
, the ring of Gaussian integers. Define , the norm of the Gaussian integer .
(where is a primitive (non-real) cube root of unity), the ring of Eisenstein integers. Define , the norm of the Eisenstein integer .
, the ring of polynomials over a field . For each nonzero polynomial , define to be the degree of .
, the ring of formal power series over the field . For each nonzero power series , define as the order of , that is the degree of the smallest power of occurring in . In particular, for two nonzero power series and , if and only if divides .
Any discrete valuation ring. Define to be the highest power of the maximal ideal containing . Equivalently, let be a generator of , and be the unique integer such that is an associate of , then define . The previous example is a special case of this.
A Dedekind domain with finitely many nonzero prime ideals . Define , where is the discrete valuation corresponding to the ideal .
Examples of domains that are not Euclidean domains include:
Every domain that is not a principal ideal domain, such as the ring of polynomials in at least two indeterminates over a field, or the ring of univariate polynomials with integer coefficients, or the number ring .
The ring of integers of , consisting of the numbers where and are integers and both even or both odd. It is a principal ideal domain that is not Euclidean.
The ring is also a principal ideal domain that is not Euclidean. To see that it is not a Euclidean domain, it suffices to show that for every non-zero prime , the map induced by the quotient map is not surjective.
Properties
Let R be a domain and f a Euclidean function on R. Then: | [
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10376_3 | R is a principal ideal domain (PID). In fact, if I is a nonzero ideal of R then any element a of I\{0} with minimal value (on that set) of f(a) is a generator of I. As a consequence R is also a unique factorization domain and a Noetherian ring. With respect to general principal ideal domains, the existence of factorizations (i.e., that R is an atomic domain) is particularly easy to prove in Euclidean domains: choosing a Euclidean function f satisfying (EF2), x cannot have any decomposition into more than f(x) nonunit factors, so starting with x and repeatedly decomposing reducible factors is bound to produce a factorization into irreducible elements.
Any element of R at which f takes its globally minimal value is invertible in R. If an f satisfying (EF2) is chosen, then the converse also holds, and f takes its minimal value exactly at the invertible elements of R.
If the Euclidean property is algorithmic, i.e., if there is a division algorithm that for given a and nonzero b produces a quotient q and remainder r with and either or , then an extended Euclidean algorithm can be defined in terms of this division operation.
If a Euclidean domain is not a field then it has an element a with the following property: any element x not divisible by a can be written as x=ay+u for some unit u and some element y. This follows by taking a to be a non-unit with f(a) as small as possible. This strange property can be used to show that some principal ideal domains are not Euclidean domains, as not all PIDs have this property. For example, for d = −19, −43, −67, −163, the ring of integers of is a PID which is Euclidean, but the cases d = −1, −2, −3, −7, −11 Euclidean. | [
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10376_4 | However, in many finite extensions of Q with trivial class group, the ring of integers is Euclidean (not necessarily with respect to the absolute value of the field norm; see below).
Assuming the extended Riemann hypothesis, if K is a finite extension of Q and the ring of integers of K is a PID with an infinite number of units, then the ring of integers is Euclidean.
In particular this applies to the case of totally real quadratic number fields with trivial class group.
In addition (and without assuming ERH), if the field K is a Galois extension of Q, has trivial class group and unit rank strictly greater than three, then the ring of integers is Euclidean.
An immediate corollary of this is that if the number field is Galois over Q, its class group is trivial and the extension has degree greater than 8 then the ring of integers is necessarily Euclidean.
Norm-Euclidean fields
Algebraic number fields K come with a canonical norm function on them: the absolute value of the field norm N that takes an algebraic element α to the product of all the conjugates of α. This norm maps the ring of integers of a number field K, say OK, to the nonnegative rational integers, so it is a candidate to be a Euclidean norm on this ring. If this norm satisfies the axioms of a Euclidean function then the number field K is called norm-Euclidean or simply Euclidean. Strictly speaking it is the ring of integers that is Euclidean since fields are trivially Euclidean domains, but the terminology is standard.
If a field is not norm-Euclidean then that does not mean the ring of integers is not Euclidean, just that the field norm does not satisfy the axioms of a Euclidean function. In fact, the rings of integers of number fields may be divided in several classes:
Those that are not principal and therefore not Euclidean, such as the integers of
Those that are principal and not Euclidean, such as the integers of
Those that are Euclidean and not norm-Euclidean, such as the integers of
Those that are norm-Euclidean, such as Gaussian integers (integers of ) | [
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] |
10376_5 | The norm-Euclidean quadratic fields have been fully classified; they are , where d takes the values
−11, −7, −3, −2, −1, 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73 .
Every Euclidean imaginary quadratic field is norm-Euclidean and is one of the five first fields in the preceding list.
See also
Valuation (algebra)
Notes
References
Ring theory
Commutative algebra
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10377_0 | In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his Elements (c. 300 BC).
It is an example of an algorithm, a step-by-step procedure for performing a calculation according to well-defined rules,
and is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations.
The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. For example, 21 is the GCD of 252 and 105 (as 252 = 21 × 12 and 105 = 21 × 5), and the same number 21 is also the GCD of 105 and 252 − 105 = 147. Since this replacement reduces the larger of the two numbers, repeating this process gives successively smaller pairs of numbers until the two numbers become equal. When that occurs, they are the GCD of the original two numbers. By reversing the steps or using the extended Euclidean algorithm, the GCD can be expressed as a linear combination of the two original numbers, that is the sum of the two numbers, each multiplied by an integer (for example, 21 = 5 × 105 + (−2) × 252). The fact that the GCD can always be expressed in this way is known as Bézout's identity.
The version of the Euclidean algorithm described above (and by Euclid) can take many subtraction steps to find the GCD when one of the given numbers is much bigger than the other. A more efficient version of the algorithm shortcuts these steps, instead replacing the larger of the two numbers by its remainder when divided by the smaller of the two (with this version, the algorithm stops when reaching a zero remainder). With this improvement, the algorithm never requires more steps than five times the number of digits (base 10) of the smaller integer. This was proven by Gabriel Lamé in 1844, and marks the beginning of computational complexity theory. Additional methods for improving the algorithm's efficiency were developed in the 20th century. | [
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10377_1 | The Euclidean algorithm has many theoretical and practical applications. It is used for reducing fractions to their simplest form and for performing division in modular arithmetic. Computations using this algorithm form part of the cryptographic protocols that are used to secure internet communications, and in methods for breaking these cryptosystems by factoring large composite numbers. The Euclidean algorithm may be used to solve Diophantine equations, such as finding numbers that satisfy multiple congruences according to the Chinese remainder theorem, to construct continued fractions, and to find accurate rational approximations to real numbers. Finally, it can be used as a basic tool for proving theorems in number theory such as Lagrange's four-square theorem and the uniqueness of prime factorizations. The original algorithm was described only for natural numbers and geometric lengths (real numbers), but the algorithm was generalized in the 19th century to other types of numbers, such as Gaussian integers and polynomials of one variable. This led to modern abstract algebraic notions such as Euclidean domains.
Background: greatest common divisor
The Euclidean algorithm calculates the greatest common divisor (GCD) of two natural numbers a and b. The greatest common divisor g is the largest natural number that divides both a and b without leaving a remainder. Synonyms for the GCD include the greatest common factor (GCF), the highest common factor (HCF), the highest common divisor (HCD), and the greatest common measure (GCM). The greatest common divisor is often written as gcd(a, b) or, more simply, as (a, b), although the latter notation is ambiguous, also used for concepts such as an ideal in the ring of integers, which is closely related to GCD.
If gcd(a, b) = 1, then a and b are said to be coprime (or relatively prime). This property does not imply that a or b are themselves prime numbers. For example, neither 6 nor 35 is a prime number, since they both have two prime factors: 6 = 2 × 3 and 35 = 5 × 7. Nevertheless, 6 and 35 are coprime. No natural number other than 1 divides both 6 and 35, since they have no prime factors in common. | [
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10377_2 | Let g = gcd(a, b). Since a and b are both multiples of g, they can be written a = mg and b = ng, and there is no larger number G > g for which this is true. The natural numbers m and n must be coprime, since any common factor could be factored out of m and n to make g greater. Thus, any other number c that divides both a and b must also divide g. The greatest common divisor g of a and b is the unique (positive) common divisor of a and b that is divisible by any other common divisor c.
The GCD can be visualized as follows. Consider a rectangular area a by b, and any common divisor c that divides both a and b exactly. The sides of the rectangle can be divided into segments of length c, which divides the rectangle into a grid of squares of side length c. The greatest common divisor g is the largest value of c for which this is possible. For illustration, a 24-by-60 rectangular area can be divided into a grid of: 1-by-1 squares, 2-by-2 squares, 3-by-3 squares, 4-by-4 squares, 6-by-6 squares or 12-by-12 squares. Therefore, 12 is the greatest common divisor of 24 and 60. A 24-by-60 rectangular area can be divided into a grid of 12-by-12 squares, with two squares along one edge (24/12 = 2) and five squares along the other (60/12 = 5). | [
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10377_3 | The GCD of two numbers a and b is the product of the prime factors shared by the two numbers, where a same prime factor can be used multiple times, but only as long as the product of these factors divides both a and b. For example, since 1386 can be factored into 2 × 3 × 3 × 7 × 11, and 3213 can be factored into 3 × 3 × 3 × 7 × 17, the greatest common divisor of 1386 and 3213 equals 63 = 3 × 3 × 7, the product of their shared prime factors. If two numbers have no prime factors in common, their greatest common divisor is 1 (obtained here as an instance of the empty product), in other words they are coprime. A key advantage of the Euclidean algorithm is that it can find the GCD efficiently without having to compute the prime factors. Factorization of large integers is believed to be a computationally very difficult problem, and the security of many widely used cryptographic protocols is based upon its infeasibility.
Another definition of the GCD is helpful in advanced mathematics, particularly ring theory. The greatest common divisor g of two nonzero numbers a and b is also their smallest positive integral linear combination, that is, the smallest positive number of the form ua + vb where u and v are integers. The set of all integral linear combinations of a and b is actually the same as the set of all multiples of g (mg, where m is an integer). In modern mathematical language, the ideal generated by a and b is the ideal generated by g alone (an ideal generated by a single element is called a principal ideal, and all ideals of the integers are principal ideals). Some properties of the GCD are in fact easier to see with this description, for instance the fact that any common divisor of a and b also divides the GCD (it divides both terms of ua + vb). The equivalence of this GCD definition with the other definitions is described below.
The GCD of three or more numbers equals the product of the prime factors common to all the numbers, but it can also be calculated by repeatedly taking the GCDs of pairs of numbers. For example,
Thus, Euclid's algorithm, which computes the GCD of two integers, suffices to calculate the GCD of arbitrarily many integers. | [
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10377_4 | Description
Procedure
The Euclidean algorithm proceeds in a series of steps such that the output of each step is used as an input for the next one. Let k be an integer that counts the steps of the algorithm, starting with zero. Thus, the initial step corresponds to k = 0, the next step corresponds to k = 1, and so on.
Each step begins with two nonnegative remainders rk−1 and rk−2. Since the algorithm ensures that the remainders decrease steadily with every step, rk−1 is less than its predecessor rk−2. The goal of the kth step is to find a quotient qk and remainder rk that satisfy the equation
and that have 0 ≤ rk < rk−1. In other words, multiples of the smaller number rk−1 are subtracted from the larger number rk−2 until the remainder rk is smaller than rk−1.
In the initial step (k = 0), the remainders r−2 and r−1 equal a and b, the numbers for which the GCD is sought. In the next step (k = 1), the remainders equal b and the remainder r0 of the initial step, and so on. Thus, the algorithm can be written as a sequence of equations
If a is smaller than b, the first step of the algorithm swaps the numbers. For example, if a < b, the initial quotient q0 equals zero, and the remainder r0 is a. Thus, rk is smaller than its predecessor rk−1 for all k ≥ 0.
Since the remainders decrease with every step but can never be negative, a remainder rN must eventually equal zero, at which point the algorithm stops. The final nonzero remainder rN−1 is the greatest common divisor of a and b. The number N cannot be infinite because there are only a finite number of nonnegative integers between the initial remainder r0 and zero. | [
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] |
10377_5 | Proof of validity
The validity of the Euclidean algorithm can be proven by a two-step argument. In the first step, the final nonzero remainder rN−1 is shown to divide both a and b. Since it is a common divisor, it must be less than or equal to the greatest common divisor g. In the second step, it is shown that any common divisor of a and b, including g, must divide rN−1; therefore, g must be less than or equal to rN−1. These two conclusions are inconsistent unless rN−1 = g.
To demonstrate that rN−1 divides both a and b (the first step), rN−1 divides its predecessor rN−2
since the final remainder rN is zero. rN−1 also divides its next predecessor rN−3
because it divides both terms on the right-hand side of the equation. Iterating the same argument, rN−1 divides all the preceding remainders, including a and b. None of the preceding remainders rN−2, rN−3, etc. divide a and b, since they leave a remainder. Since rN−1 is a common divisor of a and b, rN−1 ≤ g.
In the second step, any natural number c that divides both a and b (in other words, any common divisor of a and b) divides the remainders rk. By definition, a and b can be written as multiples of c : a = mc and b = nc, where m and n are natural numbers. Therefore, c divides the initial remainder r0, since r0 = a − q0b = mc − q0nc = (m − q0n)c. An analogous argument shows that c also divides the subsequent remainders r1, r2, etc. Therefore, the greatest common divisor g must divide rN−1, which implies that g ≤ rN−1. Since the first part of the argument showed the reverse (rN−1 ≤ g), it follows that g = rN−1. Thus, g is the greatest common divisor of all the succeeding pairs:
Worked example | [
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10377_6 | Worked example
For illustration, the Euclidean algorithm can be used to find the greatest common divisor of a = 1071 and b = 462. To begin, multiples of 462 are subtracted from 1071 until the remainder is less than 462. Two such multiples can be subtracted (q0 = 2), leaving a remainder of 147:
Then multiples of 147 are subtracted from 462 until the remainder is less than 147. Three multiples can be subtracted (q1 = 3), leaving a remainder of 21:
Then multiples of 21 are subtracted from 147 until the remainder is less than 21. Seven multiples can be subtracted (q2 = 7), leaving no remainder:
Since the last remainder is zero, the algorithm ends with 21 as the greatest common divisor of 1071 and 462. This agrees with the gcd(1071, 462) found by prime factorization above. In tabular form, the steps are:
Visualization
The Euclidean algorithm can be visualized in terms of the tiling analogy given above for the greatest common divisor. Assume that we wish to cover an a-by-b rectangle with square tiles exactly, where a is the larger of the two numbers. We first attempt to tile the rectangle using b-by-b square tiles; however, this leaves an r0-by-b residual rectangle untiled, where r0 < b. We then attempt to tile the residual rectangle with r0-by-r0 square tiles. This leaves a second residual rectangle r1-by-r0, which we attempt to tile using r1-by-r1 square tiles, and so on. The sequence ends when there is no residual rectangle, i.e., when the square tiles cover the previous residual rectangle exactly. The length of the sides of the smallest square tile is the GCD of the dimensions of the original rectangle. For example, the smallest square tile in the adjacent figure is 21-by-21 (shown in red), and 21 is the GCD of 1071 and 462, the dimensions of the original rectangle (shown in green).
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10377_7 | Euclidean division
At every step k, the Euclidean algorithm computes a quotient qk and remainder rk from two numbers rk−1 and rk−2
where the rk is non-negative and is strictly less than the absolute value of rk−1. The theorem which underlies the definition of the Euclidean division ensures that such a quotient and remainder always exist and are unique.
In Euclid's original version of the algorithm, the quotient and remainder are found by repeated subtraction; that is, rk−1 is subtracted from rk−2 repeatedly until the remainder rk is smaller than rk−1. After that rk and rk−1 are exchanged and the process is iterated. Euclidean division reduces all the steps between two exchanges into a single step, which is thus more efficient. Moreover, the quotients are not needed, thus one may replace Euclidean division by the modulo operation, which gives only the remainder. Thus the iteration of the Euclidean algorithm becomes simply
Implementations
Implementations of the algorithm may be expressed in pseudocode. For example, the division-based version may be programmed as
function gcd(a, b)
while b ≠ 0
t := b
b := a mod b
a := t
return a
At the beginning of the kth iteration, the variable b holds the latest remainder rk−1, whereas the variable a holds its predecessor, rk−2. The step b := a mod b is equivalent to the above recursion formula rk ≡ rk−2 mod rk−1. The temporary variable t holds the value of rk−1 while the next remainder rk is being calculated. At the end of the loop iteration, the variable b holds the remainder rk, whereas the variable a holds its predecessor, rk−1.
(If negative inputs are allowed, or if the mod function may return negative values, the last line must be changed into return max(a, −a).) | [
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10377_8 | In the subtraction-based version, which was Euclid's original version, the remainder calculation (b := a mod b) is replaced by repeated subtraction. Contrary to the division-based version, which works with arbitrary integers as input, the subtraction-based version supposes that the input consists of positive integers and stops when a = b:
function gcd(a, b)
while a ≠ b
if a > b
a := a − b
else
b := b − a
return a
The variables a and b alternate holding the previous remainders rk−1 and rk−2. Assume that a is larger than b at the beginning of an iteration; then a equals rk−2, since rk−2 > rk−1. During the loop iteration, a is reduced by multiples of the previous remainder b until a is smaller than b. Then a is the next remainder rk. Then b is reduced by multiples of a until it is again smaller than a, giving the next remainder rk+1, and so on.
The recursive version is based on the equality of the GCDs of successive remainders and the stopping condition gcd(rN−1, 0) = rN−1.
function gcd(a, b)
if b = 0
return a
else
return gcd(b, a mod b)
(As above, if negative inputs are allowed, or if the mod function may return negative values, the instruction "return a" must be changed into "return max(a, −a)".)
For illustration, the gcd(1071, 462) is calculated from the equivalent gcd(462, 1071 mod 462) = gcd(462, 147). The latter GCD is calculated from the gcd(147, 462 mod 147) = gcd(147, 21), which in turn is calculated from the gcd(21, 147 mod 21) = gcd(21, 0) = 21.
Method of least absolute remainders
In another version of Euclid's algorithm, the quotient at each step is increased by one if the resulting negative remainder is smaller in magnitude than the typical positive remainder. Previously, the equation | [
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10377_9 | assumed that . However, an alternative negative remainder can be computed:
if or
if .
If is replaced by when , then one gets a variant of Euclidean algorithm such that
at each step.
Leopold Kronecker has shown that this version requires the fewest steps of any version of Euclid's algorithm. More generally, it has been proven that, for every input numbers a and b, the number of steps is minimal if and only if is chosen in order that where is the golden ratio.
Historical development
The Euclidean algorithm is one of the oldest algorithms in common use. It appears in Euclid's Elements (c. 300 BC), specifically in Book 7 (Propositions 1–2) and Book 10 (Propositions 2–3). In Book 7, the algorithm is formulated for integers, whereas in Book 10, it is formulated for lengths of line segments. (In modern usage, one would say it was formulated there for real numbers. But lengths, areas, and volumes, represented as real numbers in modern usage, are not measured in the same units and there is no natural unit of length, area, or volume; the concept of real numbers was unknown at that time.) The latter algorithm is geometrical. The GCD of two lengths a and b corresponds to the greatest length g that measures a and b evenly; in other words, the lengths a and b are both integer multiples of the length g.
The algorithm was probably not discovered by Euclid, who compiled results from earlier mathematicians in his Elements. The mathematician and historian B. L. van der Waerden suggests that Book VII derives from a textbook on number theory written by mathematicians in the school of Pythagoras. The algorithm was probably known by Eudoxus of Cnidus (about 375 BC). The algorithm may even pre-date Eudoxus, judging from the use of the technical term ἀνθυφαίρεσις (anthyphairesis, reciprocal subtraction) in works by Euclid and Aristotle. | [
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10377_10 | Centuries later, Euclid's algorithm was discovered independently both in India and in China, primarily to solve Diophantine equations that arose in astronomy and making accurate calendars. In the late 5th century, the Indian mathematician and astronomer Aryabhata described the algorithm as the "pulverizer", perhaps because of its effectiveness in solving Diophantine equations. Although a special case of the Chinese remainder theorem had already been described in the Chinese book Sunzi Suanjing, the general solution was published by Qin Jiushao in his 1247 book Shushu Jiuzhang (數書九章 Mathematical Treatise in Nine Sections). The Euclidean algorithm was first described numerically and popularized in Europe in the second edition of Bachet's Problèmes plaisants et délectables (Pleasant and enjoyable problems, 1624). In Europe, it was likewise used to solve Diophantine equations and in developing continued fractions. The extended Euclidean algorithm was published by the English mathematician Nicholas Saunderson, who attributed it to Roger Cotes as a method for computing continued fractions efficiently. | [
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10377_11 | In the 19th century, the Euclidean algorithm led to the development of new number systems, such as Gaussian integers and Eisenstein integers. In 1815, Carl Gauss used the Euclidean algorithm to demonstrate unique factorization of Gaussian integers, although his work was first published in 1832. Gauss mentioned the algorithm in his Disquisitiones Arithmeticae (published 1801), but only as a method for continued fractions. Peter Gustav Lejeune Dirichlet seems to have been the first to describe the Euclidean algorithm as the basis for much of number theory. Lejeune Dirichlet noted that many results of number theory, such as unique factorization, would hold true for any other system of numbers to which the Euclidean algorithm could be applied. Lejeune Dirichlet's lectures on number theory were edited and extended by Richard Dedekind, who used Euclid's algorithm to study algebraic integers, a new general type of number. For example, Dedekind was the first to prove Fermat's two-square theorem using the unique factorization of Gaussian integers. Dedekind also defined the concept of a Euclidean domain, a number system in which a generalized version of the Euclidean algorithm can be defined (as described below). In the closing decades of the 19th century, the Euclidean algorithm gradually became eclipsed by Dedekind's more general theory of ideals.
Other applications of Euclid's algorithm were developed in the 19th century. In 1829, Charles Sturm showed that the algorithm was useful in the Sturm chain method for counting the real roots of polynomials in any given interval.
The Euclidean algorithm was the first integer relation algorithm, which is a method for finding integer relations between commensurate real numbers. Several novel integer relation algorithms have been developed, such as the algorithm of Helaman Ferguson and R.W. Forcade (1979) and the LLL algorithm. | [
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10377_12 | In 1969, Cole and Davie developed a two-player game based on the Euclidean algorithm, called The Game of Euclid, which has an optimal strategy. The players begin with two piles of a and b stones. The players take turns removing m multiples of the smaller pile from the larger. Thus, if the two piles consist of x and y stones, where x is larger than y, the next player can reduce the larger pile from x stones to x − my stones, as long as the latter is a nonnegative integer. The winner is the first player to reduce one pile to zero stones.
Mathematical applications
Bézout's identity
Bézout's identity states that the greatest common divisor g of two integers a and b can be represented as a linear sum of the original two numbers a and b. In other words, it is always possible to find integers s and t such that g = sa + tb.
The integers s and t can be calculated from the quotients q0, q1, etc. by reversing the order of equations in Euclid's algorithm. Beginning with the next-to-last equation, g can be expressed in terms of the quotient qN−1 and the two preceding remainders, rN−2 and rN−3:
Those two remainders can be likewise expressed in terms of their quotients and preceding remainders,
and
Substituting these formulae for rN−2 and rN−3 into the first equation yields g as a linear sum of the remainders rN−4 and rN−5. The process of substituting remainders by formulae involving their predecessors can be continued until the original numbers a and b are reached:
After all the remainders r0, r1, etc. have been substituted, the final equation expresses g as a linear sum of a and b: g = sa + tb. Bézout's identity, and therefore the previous algorithm, can both be generalized to the context of Euclidean domains. | [
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] |
10377_13 | Principal ideals and related problems
Bézout's identity provides yet another definition of the greatest common divisor g of two numbers a and b. Consider the set of all numbers ua + vb, where u and v are any two integers. Since a and b are both divisible by g, every number in the set is divisible by g. In other words, every number of the set is an integer multiple of g. This is true for every common divisor of a and b. However, unlike other common divisors, the greatest common divisor is a member of the set; by Bézout's identity, choosing u = s and v = t gives g. A smaller common divisor cannot be a member of the set, since every member of the set must be divisible by g. Conversely, any multiple m of g can be obtained by choosing u = ms and v = mt, where s and t are the integers of Bézout's identity. This may be seen by multiplying Bézout's identity by m,
Therefore, the set of all numbers ua + vb is equivalent to the set of multiples m of g. In other words, the set of all possible sums of integer multiples of two numbers (a and b) is equivalent to the set of multiples of gcd(a, b). The GCD is said to be the generator of the ideal of a and b. This GCD definition led to the modern abstract algebraic concepts of a principal ideal (an ideal generated by a single element) and a principal ideal domain (a domain in which every ideal is a principal ideal).
Certain problems can be solved using this result. For example, consider two measuring cups of volume a and b. By adding/subtracting u multiples of the first cup and v multiples of the second cup, any volume ua + vb can be measured out. These volumes are all multiples of g = gcd(a, b).
Extended Euclidean algorithm
The integers s and t of Bézout's identity can be computed efficiently using the extended Euclidean algorithm. This extension adds two recursive equations to Euclid's algorithm
with the starting values | [
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10377_14 | with the starting values
Using this recursion, Bézout's integers s and t are given by s = sN and t = tN, where N+1 is the step on which the algorithm terminates with rN+1 = 0.
The validity of this approach can be shown by induction. Assume that the recursion formula is correct up to step k − 1 of the algorithm; in other words, assume that
for all j less than k. The kth step of the algorithm gives the equation
Since the recursion formula has been assumed to be correct for rk−2 and rk−1, they may be expressed in terms of the corresponding s and t variables
Rearranging this equation yields the recursion formula for step k, as required
Matrix method
The integers s and t can also be found using an equivalent matrix method. The sequence of equations of Euclid's algorithm
can be written as a product of 2-by-2 quotient matrices multiplying a two-dimensional remainder vector
Let M represent the product of all the quotient matrices
This simplifies the Euclidean algorithm to the form
To express g as a linear sum of a and b, both sides of this equation can be multiplied by the inverse of the matrix M. The determinant of M equals (−1)N+1, since it equals the product of the determinants of the quotient matrices, each of which is negative one. Since the determinant of M is never zero, the vector of the final remainders can be solved using the inverse of M
Since the top equation gives
the two integers of Bézout's identity are s = (−1)N+1m22 and t = (−1)Nm12. The matrix method is as efficient as the equivalent recursion, with two multiplications and two additions per step of the Euclidean algorithm. | [
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10377_15 | Euclid's lemma and unique factorization
Bézout's identity is essential to many applications of Euclid's algorithm, such as demonstrating the unique factorization of numbers into prime factors. To illustrate this, suppose that a number L can be written as a product of two factors u and v, that is, L = uv. If another number w also divides L but is coprime with u, then w must divide v, by the following argument: If the greatest common divisor of u and w is 1, then integers s and t can be found such that
by Bézout's identity. Multiplying both sides by v gives the relation
Since w divides both terms on the right-hand side, it must also divide the left-hand side, v. This result is known as Euclid's lemma. Specifically, if a prime number divides L, then it must divide at least one factor of L. Conversely, if a number w is coprime to each of a series of numbers a1, a2, ..., an, then w is also coprime to their product, a1 × a2 × ... × an.
Euclid's lemma suffices to prove that every number has a unique factorization into prime numbers. To see this, assume the contrary, that there are two independent factorizations of L into m and n prime factors, respectively
Since each prime p divides L by assumption, it must also divide one of the q factors; since each q is prime as well, it must be that p = q. Iteratively dividing by the p factors shows that each p has an equal counterpart q; the two prime factorizations are identical except for their order. The unique factorization of numbers into primes has many applications in mathematical proofs, as shown below.
Linear Diophantine equations
Diophantine equations are equations in which the solutions are restricted to integers; they are named after the 3rd-century Alexandrian mathematician Diophantus. A typical linear Diophantine equation seeks integers x and y such that
where a, b and c are given integers. This can be written as an equation for x in modular arithmetic: | [
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10377_16 | Let g be the greatest common divisor of a and b. Both terms in ax + by are divisible by g; therefore, c must also be divisible by g, or the equation has no solutions. By dividing both sides by c/g, the equation can be reduced to Bezout's identity
where s and t can be found by the extended Euclidean algorithm. This provides one solution to the Diophantine equation, x1 = s (c/g) and y1 = t (c/g).
In general, a linear Diophantine equation has no solutions, or an infinite number of solutions. To find the latter, consider two solutions, (x1, y1) and (x2, y2), where
or equivalently
Therefore, the smallest difference between two x solutions is b/g, whereas the smallest difference between two y solutions is a/g. Thus, the solutions may be expressed as
.
By allowing u to vary over all possible integers, an infinite family of solutions can be generated from a single solution (x1, y1). If the solutions are required to be positive integers (x > 0, y > 0), only a finite number of solutions may be possible. This restriction on the acceptable solutions allows some systems of Diophantine equations with more unknowns than equations to have a finite number of solutions; this is impossible for a system of linear equations when the solutions can be any real number (see Underdetermined system). | [
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10377_17 | Multiplicative inverses and the RSA algorithm
A finite field is a set of numbers with four generalized operations. The operations are called addition, subtraction, multiplication and division and have their usual properties, such as commutativity, associativity and distributivity. An example of a finite field is the set of 13 numbers {0, 1, 2, ..., 12} using modular arithmetic. In this field, the results of any mathematical operation (addition, subtraction, multiplication, or division) is reduced modulo 13; that is, multiples of 13 are added or subtracted until the result is brought within the range 0–12. For example, the result of 5 × 7 = 35 mod 13 = 9. Such finite fields can be defined for any prime p; using more sophisticated definitions, they can also be defined for any power m of a prime p m. Finite fields are often called Galois fields, and are abbreviated as GF(p) or GF(p m).
In such a field with m numbers, every nonzero element a has a unique modular multiplicative inverse, a−1 such that This inverse can be found by solving the congruence equation ax ≡ 1 mod m, or the equivalent linear Diophantine equation
This equation can be solved by the Euclidean algorithm, as described above. Finding multiplicative inverses is an essential step in the RSA algorithm, which is widely used in electronic commerce; specifically, the equation determines the integer used to decrypt the message. Although the RSA algorithm uses rings rather than fields, the Euclidean algorithm can still be used to find a multiplicative inverse where one exists. The Euclidean algorithm also has other applications in error-correcting codes; for example, it can be used as an alternative to the Berlekamp–Massey algorithm for decoding BCH and Reed–Solomon codes, which are based on Galois fields.
Chinese remainder theorem
Euclid's algorithm can also be used to solve multiple linear Diophantine equations. Such equations arise in the Chinese remainder theorem, which describes a novel method to represent an integer x. Instead of representing an integer by its digits, it may be represented by its remainders xi modulo a set of N coprime numbers mi: | [
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10377_18 | The goal is to determine x from its N remainders xi. The solution is to combine the multiple equations into a single linear Diophantine equation with a much larger modulus M that is the product of all the individual moduli mi, and define Mi as
Thus, each Mi is the product of all the moduli except mi. The solution depends on finding N new numbers hi such that
With these numbers hi, any integer x can be reconstructed from its remainders xi by the equation
Since these numbers hi are the multiplicative inverses of the Mi, they may be found using Euclid's algorithm as described in the previous subsection.
Stern–Brocot tree
The Euclidean algorithm can be used to arrange the set of all positive rational numbers into an infinite binary search tree, called the Stern–Brocot tree.
The number 1 (expressed as a fraction 1/1) is placed at the root of the tree, and the location of any other number a/b can be found by computing gcd(a,b) using the original form of the Euclidean algorithm, in which each step replaces the larger of the two given numbers by its difference with the smaller number (not its remainder), stopping when two equal numbers are reached. A step of the Euclidean algorithm that replaces the first of the two numbers corresponds to a step in the tree from a node to its right child, and a step that replaces the second of the two numbers corresponds to a step in the tree from a node to its left child. The sequence of steps constructed in this way does not depend on whether a/b is given in lowest terms, and forms a path from the root to a node containing the number a/b. This fact can be used to prove that each positive rational number appears exactly once in this tree.
For example, 3/4 can be found by starting at the root, going to the left once, then to the right twice:
The Euclidean algorithm has almost the same relationship to another binary tree on the rational numbers called the Calkin–Wilf tree. The difference is that the path is reversed: instead of producing a path from the root of the tree to a target, it produces a path from the target to the root. | [
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] |
10377_19 | Continued fractions
The Euclidean algorithm has a close relationship with continued fractions. The sequence of equations can be written in the form
The last term on the right-hand side always equals the inverse of the left-hand side of the next equation. Thus, the first two equations may be combined to form
The third equation may be used to substitute the denominator term r1/r0, yielding
The final ratio of remainders rk/rk−1 can always be replaced using the next equation in the series, up to the final equation. The result is a continued fraction
In the worked example above, the gcd(1071, 462) was calculated, and the quotients qk were 2, 3 and 7, respectively. Therefore, the fraction 1071/462 may be written
as can be confirmed by calculation.
Factorization algorithms
Calculating a greatest common divisor is an essential step in several integer factorization algorithms, such as Pollard's rho algorithm, Shor's algorithm, Dixon's factorization method and the Lenstra elliptic curve factorization. The Euclidean algorithm may be used to find this GCD efficiently. Continued fraction factorization uses continued fractions, which are determined using Euclid's algorithm.
Algorithmic efficiency
The computational efficiency of Euclid's algorithm has been studied thoroughly. This efficiency can be described by the number of division steps the algorithm requires, multiplied by the computational expense of each step. The first known analysis of Euclid's algorithm is due to A. A. L. Reynaud in 1811, who showed that the number of division steps on input (u, v) is bounded by v; later he improved this to v/2 + 2. Later, in 1841, P. J. E. Finck showed that the number of division steps is at most 2 log2 v + 1, and hence Euclid's algorithm runs in time polynomial in the size of the input. Émile Léger, in 1837, studied the worst case, which is when the inputs are consecutive Fibonacci numbers. Finck's analysis was refined by Gabriel Lamé in 1844, who showed that the number of steps required for completion is never more than five times the number h of base-10 digits of the smaller number b. | [
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10377_20 | In the uniform cost model (suitable for analyzing the complexity of gcd calculation on numbers that fit into a single machine word), each step of the algorithm takes constant time, and Lamé's analysis implies that the total running time is also O(h). However, in a model of computation suitable for computation with larger numbers, the computational expense of a single remainder computation in the algorithm can be as large as O(h2). In this case the total time for all of the steps of the algorithm can be analyzed using a telescoping series, showing that it is also O(h2). Modern algorithmic techniques based on the Schönhage–Strassen algorithm for fast integer multiplication can be used to speed this up, leading to quasilinear algorithms for the GCD.
Number of steps
The number of steps to calculate the GCD of two natural numbers, a and b, may be denoted by T(a, b). If g is the GCD of a and b, then a = mg and b = ng for two coprime numbers m and n. Then
as may be seen by dividing all the steps in the Euclidean algorithm by g. By the same argument, the number of steps remains the same if a and b are multiplied by a common factor w: T(a, b) = T(wa, wb). Therefore, the number of steps T may vary dramatically between neighboring pairs of numbers, such as T(a, b) and T(a, b + 1), depending on the size of the two GCDs.
The recursive nature of the Euclidean algorithm gives another equation
where T(x, 0) = 0 by assumption.
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] |
10377_21 | Worst-case
If the Euclidean algorithm requires N steps for a pair of natural numbers a > b > 0, the smallest values of a and b for which this is true are the Fibonacci numbers FN+2 and FN+1, respectively. More precisely, if the Euclidean algorithm requires N steps for the pair a > b, then one has a ≥ FN+2 and b ≥ FN+1. This can be shown by induction. If N = 1, b divides a with no remainder; the smallest natural numbers for which this is true is b = 1 and a = 2, which are F2 and F3, respectively. Now assume that the result holds for all values of N up to M − 1. The first step of the M-step algorithm is a = q0b + r0, and the Euclidean algorithm requires M − 1 steps for the pair b > r0. By induction hypothesis, one has b ≥ FM+1 and r0 ≥ FM. Therefore, a = q0b + r0 ≥ b + r0 ≥ FM+1 + FM = FM+2,
which is the desired inequality.
This proof, published by Gabriel Lamé in 1844, represents the beginning of computational complexity theory, and also the first practical application of the Fibonacci numbers.
This result suffices to show that the number of steps in Euclid's algorithm can never be more than five times the number of its digits (base 10). For if the algorithm requires N steps, then b is greater than or equal to FN+1 which in turn is greater than or equal to φN−1, where φ is the golden ratio. Since b ≥ φN−1, then N − 1 ≤ logφb. Since log10φ > 1/5, (N − 1)/5 < log10φ logφb = log10b. Thus, N ≤ 5 log10b. Thus, the Euclidean algorithm always needs less than O(h) divisions, where h is the number of digits in the smaller number b.
Average
The average number of steps taken by the Euclidean algorithm has been defined in three different ways. The first definition is the average time T(a) required to calculate the GCD of a given number a and a smaller natural number b chosen with equal probability from the integers 0 to a − 1 | [
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10377_22 | However, since T(a, b) fluctuates dramatically with the GCD of the two numbers, the averaged function T(a) is likewise "noisy".
To reduce this noise, a second average τ(a) is taken over all numbers coprime with a
There are φ(a) coprime integers less than a, where φ is Euler's totient function. This tau average grows smoothly with a
with the residual error being of order a−(1/6) + ε, where ε is infinitesimal. The constant C (Porter's Constant) in this formula equals
where γ is the Euler–Mascheroni constant and ζ' is the derivative of the Riemann zeta function. The leading coefficient (12/π2) ln 2 was determined by two independent methods.
Since the first average can be calculated from the tau average by summing over the divisors d of a
it can be approximated by the formula
where Λ(d) is the Mangoldt function.
A third average Y(n) is defined as the mean number of steps required when both a and b are chosen randomly (with uniform distribution) from 1 to n
Substituting the approximate formula for T(a) into this equation yields an estimate for Y(n)
Computational expense per step
In each step k of the Euclidean algorithm, the quotient qk and remainder rk are computed for a given pair of integers rk−2 and rk−1
The computational expense per step is associated chiefly with finding qk, since the remainder rk can be calculated quickly from rk−2, rk−1, and qk
The computational expense of dividing h-bit numbers scales as O(h(ℓ+1)), where ℓ is the length of the quotient. | [
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10377_23 | For comparison, Euclid's original subtraction-based algorithm can be much slower. A single integer division is equivalent to the quotient q number of subtractions. If the ratio of a and b is very large, the quotient is large and many subtractions will be required. On the other hand, it has been shown that the quotients are very likely to be small integers. The probability of a given quotient q is approximately ln|u/(u − 1)| where u = (q + 1)2. For illustration, the probability of a quotient of 1, 2, 3, or 4 is roughly 41.5%, 17.0%, 9.3%, and 5.9%, respectively. Since the operation of subtraction is faster than division, particularly for large numbers, the subtraction-based Euclid's algorithm is competitive with the division-based version. This is exploited in the binary version of Euclid's algorithm.
Combining the estimated number of steps with the estimated computational expense per step shows that the Euclid's algorithm grows quadratically (h2) with the average number of digits h in the initial two numbers a and b. Let h0, h1, ..., hN−1 represent the number of digits in the successive remainders r0, r1, ..., rN−1. Since the number of steps N grows linearly with h, the running time is bounded by
Alternative methods
Euclid's algorithm is widely used in practice, especially for small numbers, due to its simplicity. For comparison, the efficiency of alternatives to Euclid's algorithm may be determined. | [
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10377_24 | One inefficient approach to finding the GCD of two natural numbers a and b is to calculate all their common divisors; the GCD is then the largest common divisor. The common divisors can be found by dividing both numbers by successive integers from 2 to the smaller number b. The number of steps of this approach grows linearly with b, or exponentially in the number of digits. Another inefficient approach is to find the prime factors of one or both numbers. As noted above, the GCD equals the product of the prime factors shared by the two numbers a and b. Present methods for prime factorization are also inefficient; many modern cryptography systems even rely on that inefficiency.
The binary GCD algorithm is an efficient alternative that substitutes division with faster operations by exploiting the binary representation used by computers. However, this alternative also scales like O(h²). It is generally faster than the Euclidean algorithm on real computers, even though it scales in the same way. Additional efficiency can be gleaned by examining only the leading digits of the two numbers a and b. The binary algorithm can be extended to other bases (k-ary algorithms), with up to fivefold increases in speed. Lehmer's GCD algorithm uses the same general principle as the binary algorithm to speed up GCD computations in arbitrary bases.
A recursive approach for very large integers (with more than 25,000 digits) leads to quasilinear integer GCD algorithms, such as those of Schönhage, and Stehlé and Zimmermann. These algorithms exploit the 2×2 matrix form of the Euclidean algorithm given above. These quasilinear methods generally scale as
Generalizations
Although the Euclidean algorithm is used to find the greatest common divisor of two natural numbers (positive integers), it may be generalized to the real numbers, and to other mathematical objects, such as polynomials, quadratic integers and Hurwitz quaternions. In the latter cases, the Euclidean algorithm is used to demonstrate the crucial property of unique factorization, i.e., that such numbers can be factored uniquely into irreducible elements, the counterparts of prime numbers. Unique factorization is essential to many proofs of number theory. | [
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10377_25 | Rational and real numbers
Euclid's algorithm can be applied to real numbers, as described by Euclid in Book 10 of his Elements. The goal of the algorithm is to identify a real number such that two given real numbers, and , are integer multiples of it: and , where and are integers. This identification is equivalent to finding an integer relation among the real numbers and ; that is, it determines integers and such that . Euclid uses this algorithm to treat the question of incommensurable lengths.
The real-number Euclidean algorithm differs from its integer counterpart in two respects. First, the remainders are real numbers, although the quotients are integers as before. Second, the algorithm is not guaranteed to end in a finite number of steps. If it does, the fraction is a rational number, i.e., the ratio of two integers
and can be written as a finite continued fraction . If the algorithm does not stop, the fraction is an irrational number and can be described by an infinite continued fraction . Examples of infinite continued fractions are the golden ratio and the square root of two, . The algorithm is unlikely to stop, since almost all ratios of two real numbers are irrational.
An infinite continued fraction may be truncated at a step to yield an approximation to that improves as is increased. The approximation is described by convergents ; the numerator and denominators are coprime and obey the recurrence relation
where and are the initial values of the recursion. The convergent is the best rational number approximation to with denominator :
Polynomials
Polynomials in a single variable x can be added, multiplied and factored into irreducible polynomials, which are the analogs of the prime numbers for integers. The greatest common divisor polynomial of two polynomials and is defined as the product of their shared irreducible polynomials, which can be identified using the Euclidean algorithm. The basic procedure is similar to that for integers. At each step , a quotient polynomial and a remainder polynomial are identified to satisfy the recursive equation | [
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10377_26 | where and . Each quotient polynomial is chosen such that each remainder is either zero or has a degree that is smaller than the degree of its predecessor: . Since the degree is a nonnegative integer, and since it decreases with every step, the Euclidean algorithm concludes in a finite number of steps. The last nonzero remainder is the greatest common divisor of the original two polynomials, and .
For example, consider the following two quartic polynomials, which each factor into two quadratic polynomials
Dividing by yields a remainder . In the next step, is divided by yielding a remainder . Finally, dividing by yields a zero remainder, indicating that is the greatest common divisor polynomial of and , consistent with their factorization.
Many of the applications described above for integers carry over to polynomials. The Euclidean algorithm can be used to solve linear Diophantine equations and Chinese remainder problems for polynomials; continued fractions of polynomials can also be defined.
The polynomial Euclidean algorithm has other applications, such as Sturm chains, a method for counting the zeros of a polynomial that lie inside a given real interval. This in turn has applications in several areas, such as the Routh–Hurwitz stability criterion in control theory.
Finally, the coefficients of the polynomials need not be drawn from integers, real numbers or even the complex numbers. For example, the coefficients may be drawn from a general field, such as the finite fields described above. The corresponding conclusions about the Euclidean algorithm and its applications hold even for such polynomials.
Gaussian integers
The Gaussian integers are complex numbers of the form , where and are ordinary integers and is the square root of negative one. By defining an analog of the Euclidean algorithm, Gaussian integers can be shown to be uniquely factorizable, by the argument above. This unique factorization is helpful in many applications, such as deriving all Pythagorean triples or proving Fermat's theorem on sums of two squares. In general, the Euclidean algorithm is convenient in such applications, but not essential; for example, the theorems can often be proven by other arguments.
The Euclidean algorithm developed for two Gaussian integers and is nearly the same as that for ordinary integers, but differs in two respects. As before, the task at each step is to identify a quotient and a remainder such that | [
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10377_27 | where , where , and where every remainder is strictly smaller than its predecessor: . The first difference is that the quotients and remainders are themselves Gaussian integers, and thus are complex numbers. The quotients are generally found by rounding the real and complex parts of the exact ratio (such as the complex number ) to the nearest integers. The second difference lies in the necessity of defining how one complex remainder can be "smaller" than another. To do this, a norm function is defined, which converts every Gaussian integer into an ordinary integer. After each step of the Euclidean algorithm, the norm of the remainder is smaller than the norm of the preceding remainder, . Since the norm is a nonnegative integer and decreases with every step, the Euclidean algorithm for Gaussian integers ends in a finite number of steps. The final nonzero remainder is , the Gaussian integer of largest norm that divides both and ; it is unique up to multiplication by a unit, or .
Many of the other applications of the Euclidean algorithm carry over to Gaussian integers. For example, it can be used to solve linear Diophantine equations and Chinese remainder problems for Gaussian integers; continued fractions of Gaussian integers can also be defined.
Euclidean domains
A set of elements under two binary operations, denoted as addition and multiplication, is called a Euclidean domain if it forms a commutative ring and, roughly speaking, if a generalized Euclidean algorithm can be performed on them. The two operations of such a ring need not be the addition and multiplication of ordinary arithmetic; rather, they can be more general, such as the operations of a mathematical group or monoid. Nevertheless, these general operations should respect many of the laws governing ordinary arithmetic, such as commutativity, associativity and distributivity. | [
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10377_28 | The generalized Euclidean algorithm requires a Euclidean function, i.e., a mapping from into the set of nonnegative integers such that, for any two nonzero elements and in , there exist and in such that and . Examples of such mappings are the absolute value for integers, the degree for univariate polynomials, and the norm for Gaussian integers above. The basic principle is that each step of the algorithm reduces f inexorably; hence, if can be reduced only a finite number of times, the algorithm must stop in a finite number of steps. This principle relies on the well-ordering property of the non-negative integers, which asserts that every non-empty set of non-negative integers has a smallest member.
The fundamental theorem of arithmetic applies to any Euclidean domain: Any number from a Euclidean domain can be factored uniquely into irreducible elements. Any Euclidean domain is a unique factorization domain (UFD), although the converse is not true. The Euclidean domains and the UFD's are subclasses of the GCD domains, domains in which a greatest common divisor of two numbers always exists. In other words, a greatest common divisor may exist (for all pairs of elements in a domain), although it may not be possible to find it using a Euclidean algorithm. A Euclidean domain is always a principal ideal domain (PID), an integral domain in which every ideal is a principal ideal. Again, the converse is not true: not every PID is a Euclidean domain. | [
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10377_29 | The unique factorization of Euclidean domains is useful in many applications. For example, the unique factorization of the Gaussian integers is convenient in deriving formulae for all Pythagorean triples and in proving Fermat's theorem on sums of two squares. Unique factorization was also a key element in an attempted proof of Fermat's Last Theorem published in 1847 by Gabriel Lamé, the same mathematician who analyzed the efficiency of Euclid's algorithm, based on a suggestion of Joseph Liouville. Lamé's approach required the unique factorization of numbers of the form , where and are integers, and is an th root of 1, that is, . Although this approach succeeds for some values of (such as , the Eisenstein integers), in general such numbers do factor uniquely. This failure of unique factorization in some cyclotomic fields led Ernst Kummer to the concept of ideal numbers and, later, Richard Dedekind to ideals.
Unique factorization of quadratic integers
The quadratic integer rings are helpful to illustrate Euclidean domains. Quadratic integers are generalizations of the Gaussian integers in which the imaginary unit i is replaced by a number . Thus, they have the form , where and are integers and has one of two forms, depending on a parameter . If does not equal a multiple of four plus one, then
If, however, D does equal a multiple of four plus one, then
If the function corresponds to a norm function, such as that used to order the Gaussian integers above, then the domain is known as norm-Euclidean. The norm-Euclidean rings of quadratic integers are exactly those where is one of the values −11, −7, −3, −2, −1, 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, or 73. The cases and yield the Gaussian integers and Eisenstein integers, respectively. | [
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10377_30 | If is allowed to be any Euclidean function, then the list of possible values of for which the domain is Euclidean is not yet known. The first example of a Euclidean domain that was not norm-Euclidean (with ) was published in 1994. In 1973, Weinberger proved that a quadratic integer ring with is Euclidean if, and only if, it is a principal ideal domain, provided that the generalized Riemann hypothesis holds.
Noncommutative rings
The Euclidean algorithm may be applied to some noncommutative rings such as the set of Hurwitz quaternions. Let and represent two elements from such a ring. They have a common right divisor if and for some choice of and in the ring. Similarly, they have a common left divisor if and for some choice of and in the ring. Since multiplication is not commutative, there are two versions of the Euclidean algorithm, one for right divisors and one for left divisors. Choosing the right divisors, the first step in finding the by the Euclidean algorithm can be written
where represents the quotient and the remainder. This equation shows that any common right divisor of and is likewise a common divisor of the remainder . The analogous equation for the left divisors would be
With either choice, the process is repeated as above until the greatest common right or left divisor is identified. As in the Euclidean domain, the "size" of the remainder (formally, its norm) must be strictly smaller than , and there must be only a finite number of possible sizes for , so that the algorithm is guaranteed to terminate.
Most of the results for the GCD carry over to noncommutative numbers. For example, Bézout's identity states that the right can be expressed as a linear combination of and . In other words, there are numbers and such that
The analogous identity for the left GCD is nearly the same:
Bézout's identity can be used to solve Diophantine equations. For instance, one of the standard proofs of Lagrange's four-square theorem, that every positive integer can be represented as a sum of four squares, is based on quaternion GCDs in this way.
See also
Euclidean rhythm, a method for using the Euclidean algorithm to generate musical rhythms
Notes | [
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] |
10377_31 | Notes
References
Bibliography
. See also Vorlesungen über Zahlentheorie
External links
Demonstrations of Euclid's algorithm
Euclid's Algorithm at cut-the-knot
The Euclidean Algorithm at MathPages
Euclid's Game at cut-the-knot
Music and Euclid's algorithm
Number theoretic algorithms
Articles with example pseudocode
Articles containing proofs
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10378_0 | The European Centre for Medium-Range Weather Forecasts (ECMWF) is an independent intergovernmental organisation supported by most of the nations of Europe. It is based at three sites: Shinfield Park, Reading, United Kingdom; Bologna, Italy; and Bonn, Germany. It operates one of the largest supercomputer complexes in Europe and the world's largest archive of numerical weather prediction data.
History
ECMWF was established in 1975, in recognition of the need to pool the scientific and technical resources of Europe's meteorological services and institutions for the production of weather forecasts for medium-range timescales (up to approximately two weeks) and of the economic and social benefits expected from it. The Centre employs about 350 staff, mostly appointed from across the member states and co-operating states.
In 2017, the centre's member states accepted an offer from the Italian Government to move ECMWF's data centre to Bologna, Italy. The new site, a former tobacco factory, would be redesigned by the architecture firm gmp.
During 2020, the Centre arranged to move its Copernicus operations away from Reading and into European Union territory. Following bids from Toulouse, Italy, Austria, Germany, Spain and Ireland, eventually Bonn (Germany) was chosen. The move has been directly attributed to Brexit.
Objectives
ECMWF aims to provide accurate medium-range global weather forecasts out to 15 days and seasonal forecasts out to 12 months. Its products are provided to the national weather services of its member states and co-operating states as a complement to their national short-range and climatological activities, and those national states use ECMWF's products for their own national duties, in particular to give early warning of potentially damaging severe weather.
ECMWF's core mission is to:
Produce numerical weather forecasts and monitor planetary systems that influence weather
Carry out scientific and technical research to improve forecast skill
Maintain an archive of meteorological data
To deliver this core mission, the Centre provides:
Twice-daily global numerical weather forecasts
Air quality analysis
Atmospheric composition monitoring
Climate monitoring
Ocean circulation analysis
Hydrological prediction | [
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10378_1 | The Centre develops and operates global atmospheric models and data assimilation systems for the dynamics, thermodynamics and composition of the Earth's atmosphere and for interacting parts of the Earth-system. It uses numerical weather prediction methods to prepare forecasts and their initial conditions, and it contributes to monitoring the relevant parts of the Earth system.
Work and projects
Forecasting
Numerical weather prediction (NWP) requires input of meteorological data, collected by satellites and earth observation systems such as automatic and manned stations, aircraft, ships and weather balloons. Assimilation of this data is used to produce an initial state of a computer model of the atmosphere, from which an atmospheric model is used to forecast the weather. These forecasts are typically:
medium-range forecasts, predicting the weather up to 15 days ahead
monthly forecasts, predicting the weather on a weekly basis 30 days ahead
seasonal forecasts up to 12 months ahead.
Over the past three decades ECMWF's wide-ranging programme of research has played a major role in developing such assimilation and modelling systems. This improves the accuracy and reliability of weather forecasting by about a day per decade, so that a seven-day forecast now (2015) is as accurate as a three-day forecast was four decades ago (1975).
Monthly and seasonal forecasts
ECMWF's monthly and seasonal forecasts provide early predictions of events such as heat waves, cold spells and droughts, as well as their impacts on sectors such as agriculture, energy and health. Since ECMWF runs a wave model, there are also predictions of coastal waves and storm surges in European waters which can be used to provide warnings.
Early warning of severe weather events
Forecasts of severe weather events allow appropriate mitigating action to be taken and contingency plans to be put into place by the authorities and the public. The increased time gained by issuing accurate warnings can save lives, for instance by evacuating people from a storm surge area. Authorities and businesses can plan to maintain services around threats such as high winds, floods or snow.
In October 2012 the ECMWF model suggested seven days in advance that Hurricane Sandy was likely to make landfall on the East Coast of the United States.
It also predicted the intensity and track of the November 2012 nor'easter, which impacted the east coast a week after Sandy. | [
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10378_2 | ECMWF's Extreme Forecast Index (EFI) was developed as a tool to identify where the EPS (Ensemble Prediction System) forecast distribution differs substantially from that of the model climate. It contains information regarding variability of weather parameters, in location and time and can highlight an abnormality of a weather situation without having to define specific space- and time-dependent thresholds.
Satellite data
ECMWF, through its partnerships with EUMETSAT, ESA, the EU and others, exploits satellite data for operational numerical weather prediction and operational seasonal forecasting with coupled atmosphere–ocean–land models. The increasing amount of satellite data and the development of more sophisticated ways of extracting information from that data have made a major contribution to improving the accuracy and utility of NWP forecasts. ECMWF continuously endeavours to improve the use of satellite observations for NWP.
Reanalysis
ECMWF supports research on climate variability using an approach known as reanalysis. This involves feeding weather observations collected over decades into a NWP system to recreate past atmospheric, sea- and land-surface conditions over specific time periods to obtain a clearer picture of how the climate has changed. Reanalysis provides a four-dimensional picture of the atmosphere and effectively allows monitoring of the variability and change of global climate, thereby contributing also to the understanding and attribution of climate change.
To date, and with support from Europe's National Meteorological Services and the European Commission, ECMWF has conducted several major reanalyses of the global atmosphere: the first ECMWF re-analysis (ERA-15) project generated reanalyses from December 1978 to February 1994; the ERA-40 project generated reanalyses from September 1957 to August 2002. The ERA-Interim reanalysis covered the period from 1979 onwards. A reanalysis product (ERA5) with higher spatial resolution (31 km) was released by ECMWF in 2019 as part of the Copernicus Climate Change Service.
Operational forecast model
ECMWF's operational forecasts are produced from its "Integrated Forecast System" (sometimes informally known in the United States as the "European model") which is run every twelve hours and forecasts out to ten days. | [
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] |
10378_3 | It includes both a "deterministic forecast" mode and an ensemble. The deterministic forecast is a single model run that is relatively high in resolution as well as in computational expense. The ensemble is relatively low (about half that of the deterministic) in resolution (and in computational expense), so less accurate. But it is run 51 times in parallel, from slightly different initial conditions to give a spread of likelihood over the range of the forecast.
As of 2021, the ECMWF's weather model is generally considered to be the most accurate weather forecasting model.
Copernicus
The centre currently serves as the Entrusted Entity responsible for delivery of two of the Services of the EU's Copernicus Programme. The two services are the Copernicus Atmosphere Monitoring Service (CAMS) and the Copernicus Climate Change Service (C3S).
The Centre arranged to move its Copernicus operations away from Reading and into Bonn (Germany). The move has been directly attributed to Brexit.
Member and co-operating states
ECMWF comprises 23 European countries:
the eighteen founding states of 1975: Austria, Belgium, Denmark, Finland, France, Germany, Greece, Republic of Ireland, Italy, Luxembourg, Netherlands, Norway, Portugal, Spain, Sweden, Switzerland, Turkey, United Kingdom.
five states that joined since 2010: Iceland (April 2011), Slovenia (December 2012), Serbia (January 2015), Croatia (January 2016), Estonia (December 2020)
It also has co-operation agreements with other states: Bulgaria, Czech Republic, Hungary, Israel, Latvia, Lithuania, North Macedonia, Montenegro, Morocco, Romania and Slovakia.
See also
EUMETNET
EUMETSAT
Copernicus Programme
References
Further reading
External links
ECMWF Re-analysis (ERA)
ECMWF on TOP500
1975 establishments in the United Kingdom
Climate of Europe
Climate of the United Kingdom
European research networks
Government agencies established in 1975
Governmental meteorological agencies in Europe
Information technology organisations based in the United Kingdom
Meteorological organizations
Organisations based in Reading, Berkshire
Research institutes established in 1975
Research institutes in Berkshire
Borough of Wokingham | [
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10380_0 | The European Broadcasting Union (EBU; , UER) is an alliance of public service media organisations, established on 12 February 1950. The organisation is made up of 112 member organisations in 55 countries, and 31 associate members from a further 20 countries. It is best known for producing the Eurovision Song Contest. It also hosted debates between candidates for the European Commission presidency for the 2014 and 2019 parliamentary elections but is unrelated to the European Union itself.
General description
EBU members are public service media (PSM) broadcasters whose output is made, financed and controlled by the public, for the public. PSM broadcasters are often established by law but are non-partisan, independent and run for the benefit of society as a whole.
EBU members come from as far north as Iceland and as far south as Egypt, from Ireland in the west and Azerbaijan in the east, and almost every nation from geographical Europe in between. Associate Members are from countries and territories beyond Europe, such as Canada, Japan, India and China. Associate members from the United States include ABC, CBS, NBC, CPB, NPR, APM and the only individual station, Chicago-based classical music radio WFMT.
Membership is for media organisations whose countries are within the European Broadcasting Area (EBA), as defined by the International Telecommunication Union, or who are members of the Council of Europe.
Members benefit from:
Access to world-class content ranging from exclusive sports rights to exchanges for news, music and children's programs.
A voice in Brussels and on international platforms lobbying for PSM and ensuring the optimal legal and technical framework.
Opportunities for sharing, learning and collaborating through conferences, working groups, training, and dedicated advice and guidance.
A centre for learning and sharing new technology and innovation with a team of experts providing strategic advice and guidance.
The EBU's highest-profile production is the Eurovision Song Contest. The EBU also organises the Eurovision Dance Contest, the Junior Eurovision Song Contest, the Eurovision Young Dancers competition, and other competitions which are modeled along similar lines.
Radio collaborations include Euroclassic Notturno – an overnight classical music stream, produced by BBC Radio 3 and broadcast in the United Kingdom as Through the Night – and special theme days, such as the annual Christmas music relays from around Europe. The EBU is a member of the International Music Council. | [
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10380_1 | Most EBU broadcasters have group deals to carry major sporting events including the FIFA World Cup and the inaugural European Championships. Another annually recurring event which is broadcast across Europe through the EBU is the Vienna New Year's Concert.
Eurovision Media Services is the business arm of the EBU and provides media services for many media organisations and sports federations around the world.
The theme music played before EBU broadcasts is Marc-Antoine Charpentier's Prelude to Te Deum. It is well known to Europeans as it is played before and after the Eurovision Song Contest and other important events.
History
The EBU was a successor to the International Broadcasting Union (IBU) that was founded in 1925 and had its administrative headquarters in Geneva and technical office in Brussels. It fostered programming exchanges between members and mediated technical disputes between members that were mostly concerned with frequency and interference issues. It was in effect taken over by Nazi Germany during the Second World War and when the conflict ended in the eyes of the Allies it was a compromised organisation that they could not trust.
In the spring of 1946, representatives of the Soviet radio committee proposed forming a new organisation; however, at the same time preparations were being made for an inter-governmental “European Broadcasting Conference” (EBC) in Copenhagen in 1948 to draw up a new plan for frequency use in the European Broadcasting Area (EBA). It was considered necessary to have an organisation that could implement the “Copenhagen Wavelength Plan” but there was disagreement among broadcasters and particularly a fear expressed by the BBC that a new association might be dominated by the USSR and its proposal to give each of its constituent states one vote. France proposed that it would have four votes with the inclusion of its North African colonies. The United Kingdom felt it would have little influence with just one vote. | [
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10380_2 | On 27 June 1946, the alternative International Broadcasting Organisation (IBO) was founded with 26 members and without British participation. The following day the IBU met in General Assembly and an attempt was made to dissolve it but failed; though 18 of its 28 members left to join the IBO. For a period of time in the late 1940s both the IBU and IBO vied for the role of organising frequencies but Britain decided to be in involved in neither. The BBC attempted but failed to find suitable working arrangements with them. However, for practical purposes, the IBO rented the IBU technical centre in Brussels and employed its staff. The BBC then proposed a new solution based on the IBO changing its constitution so there will be only one member per ITU country, thus ensuring a Western majority over the USSR and its satellite states. In August 1949 a meeting took place in Stresa, Italy but it resulted in disagreement between delegates on how to resolve the problems. One proposal was for the European Broadcasting Area to be replaced by one that would exclude Eastern Europe, the Levant and North Africa.
After Stresa, a consensus emerged among the Western Europeans to form a new organisation and the BBC proposed it be based in London. Meetings in Paris on 31 October and 1 November 1949 sealed the fate of the IBU and IBO, but it was decided not to allow West Germany to be a founder of the new organisation. On 13 February 1950 the European Broadcasting Union had its first meeting with 23 members from the ITU defined European Broadcasting Area at the Imperial Hotel in Torquay, England, United Kingdom. The first president was Ian Jacob of the BBC who remained at the helm for 10 years while its operation was largely dominated by the BBC due to its financial, technical and staff input. The most important difference between the EBU and its predecessors was that EBU membership was for broadcasters and not governments. Early delegates said EBU meetings were cordial and professional and very different from the abrupt tone of its predecessors. West Germany was admitted in 1951 and a working relationship forged with the USSR's Organisation for International Radio and TV (OIRT) which existed in parallel with the EBU until its merger on 1 January 1993.
In 1967, the first concert in the International Concert Season of the European Broadcasting Union was broadcast from the Queen Elizabeth Hall in London. | [
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10380_3 | Technical activities
The objective of the EBU's technical activities is simply to assist EBU Members (see below) in this period of unprecedented technological changes. This includes the provision of technical information to Members via conferences and workshops, as well as in written form (such as the EBU Technical Review, and the EBU tech-i magazine).
The EBU also encourages active collaboration between its Members on the basis that they can freely share their knowledge and experience, thus achieving considerably more than individual Members could achieve by themselves. Much of this collaboration is achieved through Project Groups which study specific technical issues of common interest: for example, EBU Members have long been preparing for the revision of the 1961 Stockholm Plan.
The EBU places great emphasis on the use of open standards. Widespread use of open standards (such as MPEG-2, DAB, DVB, etc.) ensures interoperability between products from different vendors, as well as facilitating the exchange of programme material between EBU Members and promoting "horizontal markets" for the benefit of all consumers.
EBU Members and the EBU Technical Department have long played an important role in the development of many systems used in radio and television broadcasting, such as:
The AES/EBU digital audio interface, formally known as AES3;
Serial and parallel interfaces for digital video (ITU-R Recommendations 601 and 656);
RDS – the radio data system used on FM broadcasting.
The EBU Loudness Recommendation R 128 and 'EBU Mode' meters (EBU Tech 3341)
The EBU has also actively encouraged the development and implementation of:
Digital radio (DAB) through Eureka Project 147 and the WorldDAB Forum.
Digital Video Broadcasting (DVB) through the DVB Project and DigiTAG.
Digital radio in the bands currently used for AM broadcasting through DRM (Digital Radio Mondiale).
Standardisation of PVR systems through the TV-Anytime Forum.
Development of other content distribution networks on the internet through P2PTV; EBU Project Group D/P2P, from November 2007 to April 2008, with a trial of selected member channels, thanks to Octoshape's distribution platform. The EBU is also part of the European P2P-Next project.
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] |
10380_4 | Controversies
Greek state broadcaster (2013)
On 11 June 2013, the Greek government shut down the state broadcaster ERT, at short notice, citing government spending concerns related to the Euro crisis. In response, the European Broadcasting Union set up a makeshift studio on the same day, near the former ERT offices in Athens, in order to continue providing EBU members with the news-gathering and broadcast relay services which had formerly been provided by ERT.
The EBU put out a statement expressing its "profound dismay" at the shutdown, urged the Greek Prime Minister "to use all his powers to immediately reverse this decision" and offered the "advice, assistance and expertise necessary for ERT to be preserved".
Starting on 4 May 2014, the New Hellenic Radio, Internet and Television broadcaster began nationwide transmissions, taking over ERT's vacant active membership slot in the EBU. On 11 June 2015, two years after ERT's closure, NERIT SA renamed as ERT SA which reopened with a comprehensive program in all radio stations (with nineteen regional, two world-range and five pan-Hellenic range radio stations) and three TV channels ERT1, ERT2 and ERT3.
Belarusian state broadcaster (2021)
The Belarusian Television and Radio Company (BTRC) has been accused of repressing its own employees, having fired more than 100 people since a wave of anti-Lukashenko protests in 2020 following alleged election fraud. Many of them have also been jailed. Many voices have been raised against the participation of Belarus and the BTRC in the otherwise unpolitical Eurovision Song Contest in 2021, the argument being that the EBU would make a political statement if it did endorse Belarus by essentially and silently saying that democracy is unimportant and so are basic human rights such as freedom of speech.
On 28 May 2021, the EBU suspended the BTRC's membership as they had been "particularly alarmed by the broadcast of interviews apparently obtained under duress." BTRC was given two weeks to respond before the suspension came into effect, but did not do so publicly. The broadcaster was completely expelled from the EBU on 1 July 2021 for a period of three years. | [
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] |
10380_5 | Russian broadcasters (2022)
The three Russian members of the EBU, Channel One Russia, VGTRK, and Radio Dom Ostankino are all controlled by the Russian government. In February 2022, the Russian government recognized the independence of the Donetsk and Luhansk People's Republics, disputed territories that are internationally recognized as part of Ukraine. Ukraine's public broadcaster Suspilne called on the EBU to terminate the membership of Channel One Russia and VGTRK, and consider suspending Russia from the 2022 Eurovision Song Contest, citing the Russian governments use of both outlets to spread disinformation surrounding the Russian-Ukraine conflict. Following the 2022 Russian invasion of Ukraine, several other public broadcasters joined Suspilne in calling for Russia's exclusion from the 2022 Contest; Finland's Yle stated that they would not send a representative if Russia was allowed to participate. After initially stating that both Russia and Ukraine would be allowed to compete, the EBU announced that it would bar Russia from participating in the Contest.
The three Russian broadcasters stated that they would leave the EBU on 26 February, citing increased politicization of the organisation. The EBU released a statement saying that it was aware of the reports, but that it had not received any formal confirmation.
Members
The Member list , comprises the following 66 broadcasting companies from 55 countries.
Current members
Suspended members
Past members
Associate members
Any group or organisation from an International Telecommunication Union (ITU) member country, which provides a radio or television service outside of the European Broadcasting Area, is permitted to submit applications to the EBU for Associate Membership.
It is also noted by the EBU that any country that is granted Associate Member status does not gain access into Eurovision events with the notable exceptions of Australia, who have participated in the Eurovision Song Contest and the Junior Eurovision Song Contest since 2015, Canada in Eurovision Young Dancers between 1987 and 1989 and Kazakhstan, who have participated in Junior Eurovision since 2018, all of which were individually invited.
The list of Associate Members of EBU comprised the following 31 broadcasting companies from 20 countries .
Past associate members
The list of past associate members of EBU comprises the following 29 broadcasting companies from 18 countries and 1 autonomous territory. | [
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10380_6 | Approved participant members
Any groups or organisations from a country with International Telecommunication Union (ITU) membership, which does not qualify for either the EBU's Active or Associate memberships, but still provide a broadcasting activity for the EBU, are granted a unique Approved Participants membership, which lasts approximately five years. An application for this status may be submitted to the EBU at any given time, providing an annual fee is paid.
The following seven EBU broadcast members had status as Approved Participants in May 2016.
The following members previously had status as Approved Participants.
Organised events
The EBU in co-operation with the respective host broadcaster organises competitions and events in which its members can participate if they wish to do so. These include:
Eurovision Song Contest
The Eurovision Song Contest () is an annual international song competition between EBU Members, that was first held in Lugano, Switzerland, on 24 May 1956. Seven countries participated – each submitting two songs, for a total of 14. This was the only contest in which more than one song per country was performed: since 1957 all contests have allowed one entry per country. The 1956 contest was won by the host nation, Switzerland. The most recent host city was Rotterdam, the Netherlands, where Italy won the competition.
Let the Peoples Sing
Let the Peoples Sing is a biennial choir competition, the participants of which are chosen from radio recordings entered by EBU radio members. The final, encompassing three categories and around ten choirs, is offered as a live broadcast to all EBU members. The overall winner is awarded the Silver Rose Bowl.
Jeux sans frontières
Jeux sans frontières (, or Games Without Borders) was a Europe-wide television game show. In its original conception, it was broadcast from 1965 to 1999 under the auspices of the EBU. The original series run ended in 1982 but was revived in 1988 with a different complexion of nations and was hosted by smaller broadcasters.
Eurovision Young Musicians
Eurovision Young Musicians is a competition for European musicians that are between the ages of 12 and 21 years old. It is organised by the EBU and is a member of EMCY. The first competition was held in Manchester, the United Kingdom on 11 May 1982. | [
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10380_7 | The televised competition is held every two years, with some countries holding national heats. Since its foundation in 1982, the Eurovision Young Musicians competition has become one of the most important music competitions on an international level.
Eurovision Young Dancers
The Eurovision Young Dancers is a biennial dance showcase broadcast on television throughout Europe. The first competition was held in Reggio Emilia, Italy on 16 June 1985.
It uses a format similar to the Eurovision Song Contest, every country that is a member of the EBU has had the opportunity to send a dance act to compete for the title of "Eurovision Young Dancer". The competition is for solo dancers and all contestants must be between the ages of 16 and 21 years and not professionally engaged.
Euroclassic Notturno
Euroclassic Notturno is a six-hour sequence of classical music recordings assembled by BBC Radio from material supplied by members of the EBU and streamed back to those broadcasters by satellite for use in their overnight classical-music schedules. The recordings used are taken not from commercial CDs but from earlier (usually live) radio broadcasts.
Junior Eurovision Song Contest
Junior Eurovision Song Contest (), is an annual international song competition, that was first held in Copenhagen, Denmark, on 15 November 2003. Sixteen countries participated – each submitting one song, for a total of 16 entries. The 2003 Contest was won by Croatia. Winner of most recent contest, which took place in Paris, France, is Armenia.
Eurovision Dance Contest
The Eurovision Dance Contest (not to be confused with the Eurovision Young Dancers Competition) was an international dancing competition that was held for the first time in London, the United Kingdom on 1 September 2007. The competition was repeated in 2008 when it was held in Glasgow, United Kingdom, but has not been held since.
Eurovision Magic Circus Show
The Eurovision Magic Circus Show was an entertainment show organised by the EBU, which took place in 2010, 2011 and 2012 in Geneva. Children aged between 7-14 representing 8 countries within the EBU membership area, performed a variety of circus acts at the Geneva Christmas Circus (). The main show was also accompanied by the Magic Circus Show Orchestra.
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10380_8 | Eurovision Choir
The inaugural Eurovision Choir featuring non-professional choirs selected by EBU Members, took place on 22 July 2017 in Riga, hosted by the Latvian broadcaster Latvijas Televīzija (LTV). 9 countries took part in the first edition. Carmen Manet from Slovenia was the first winner.
European Sports Championships
The European Sports Championships is a multi-sport event involving some of the leading sports in Europe. The European Governing Bodies for athletics, aquatics, cycling, rowing, golf, gymnastics and triathlon, will coordinate their individual championships as part of the first edition in the summer of 2018, hosted by the cities of Berlin (already chosen as the host for the 2018 European Athletics Championships) and Glasgow (already chosen as the host for the 2018 European Aquatics Championships, and which will now also host the events of the other sports).
See also
Asia-Pacific Broadcasting Union
Public Broadcasting System
Caribbean Broadcasting Union
Commonwealth Broadcasting Association
Commonwealth Press Union
Europe by Satellite
International Telecommunication Union
North American Broadcasters Association
Organización de Telecomunicaciones de Iberoamérica
Notes
References
External links
EBU website
1950 establishments in Europe
Organizations established in 1950
Organisations based in Geneva
Publicly funded broadcasters | [
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10382_0 | Electrothermal-chemical (ETC) technology is an attempt to increase accuracy and muzzle energy of future tank, artillery, and close-in weapon system guns by improving the predictability and rate of expansion of propellants inside the barrel.
An electrothermal-chemical gun uses a plasma cartridge to ignite and control the ammunition's propellant, using electrical energy to trigger the process. ETC increases the performance of conventional solid propellants, reduces the effect of temperature on propellant expansion and allows for more advanced, higher density propellants to be used.
The technology has been under development since the mid-1980s and at present is actively being researched in the United States by the Army Research Laboratory, Sandia National Laboratories and defense industry contractors, including FMC Corporation, General Dynamics Land Systems, Olin Ordnance, and Soreq Nuclear Research Center. It is possible that electrothermal-chemical gun propulsion will be an integral part of US Army's future combat system and those of other countries such as Germany and the United Kingdom.
Electrothermal-chemical technology is part of a broad research and development program that encompasses all electric gun technology, such as railguns and coil guns.
Background
The constant battle between armour and armor-piercing round has led to continuous development of the main battle tank design. The evolution of American anti-tank weapons can be traced back to requirements to combat Soviet tanks. In the late 1980s, it was thought that the protection level of the Future Soviet Tank (FST) could exceed 700 mm of rolled homogeneous armour equivalence at its maximum thickness, which was effectively immune against the contemporary M829 armour piercing fin stabilized discarding sabot. In the 1980s the most immediate method available to NATO to counter Soviet advances in armour technology was the adoption of a 140 mm main gun, but this required a redesigned turret that could incorporate the larger breech and ammunition, and it also required some sort of automatic loader. Although the 140 mm gun was considered a real interim solution it was decided after the fall of the Soviet Union that the increase in muzzle energy it provided was not worth the increase in weight. Resources were therefore spent on research into other programs that could provide the needed muzzle energy. One of the most successful alternative technologies remains electrothermal-chemical ignition. | [
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10382_1 | Most proposed advances in gun technology are based on the assumption that the solid propellant as a stand-alone propulsion system is no longer capable of delivering the required muzzle energy. This requirement has been underscored by the appearance of the Russian T-90 main battle tank. Even the elongation of current gun tubes, such as the new German 120 mm L/55, which was introduced by Rheinmetall is considered only an interim solution as it does not offer the required increase in muzzle velocity. Even advanced kinetic energy ammunition such as the United States' M829A3 is considered only an interim solution against future threats. To that extent the solid propellant is considered to have reached the end of its usefulness, although it will remain the principal propulsion method for at least the next decade until newer technologies mature. To improve on the capabilities of a solid propellant weapon the electrothermal-chemical gun may see production as early as 2016.
ETC technology offers a medium-risk upgrade and is developed to the point that further improvements are so minor that it can be considered mature. The lightweight American 120 mm XM291 came close to achieving 17 MJ of muzzle energy, which is the lower-end muzzle energy spectrum for a 140 mm gun. However, the success of the XM291 does not imply the success of ETC technology as there are key parts of the propulsion system that are not yet understood or fully developed, such as the plasma ignition process. Nevertheless, there is substantial existing evidence that ETC technology is viable and worth the money to continue development. Furthermore, it can be integrated into current gun systems.
How it works | [
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10382_2 | How it works
An electrothermal-chemical gun uses a plasma cartridge to ignite and control the ammunition's propellant, using electrical energy as a catalyst to begin the process. Originally researched by Dr. Jon Parmentola for the U.S. Army, it has grown into a very plausible successor to a standard solid propellant tank gun. Since the beginning of research the United States has funded the XM291 gun project with USD 4,000,000, basic research with USD 300,000, and applied research with USD 600,000. Since then it has been proven to work, although efficiency to the level required has not yet been accomplished. ETC increases the performance of conventional solid propellants, reduces the effect of temperature on propellant expansion and allows for more advanced, higher density propellants to be used. It will also reduce pressure placed on the barrel in comparison to alternative technologies that offer the same muzzle energy given the fact that it helps spread the propellant's gas much more smoothly during ignition. Currently, there are two principal methods of plasma initiation: the flashboard large area emitter (FLARE) and the triple coaxial plasma igniter (TCPI).
Flashboard large area emitter
Flashboards run in several parallel strings to provide a large area of plasma or ultraviolet radiation and uses the breakdown and vaporization of gaps of diamonds to produce the required plasma. These parallel strings are mounted in tubes and oriented to have their gaps azimuthal to the tube's axis. It discharges by using high pressure air to move air out of the way. FLARE initiators can ignite propellants through the release of plasma, or even through the use of ultraviolet heat radiation. The absorption length of a solid propellant is sufficient to be ignited by radiation from a plasma source. However, FLARE has most likely not reached optimal design requirements and further understanding of FLARE and how it works is completely necessary to ensure the evolution of the technology. If FLARE provided the XM291 gun project with the sufficient radiative heat to ignite the propellant to achieve a muzzle energy of 17 MJ one could only imagine the possibilities with a fully developed FLARE plasma igniter. Current areas of study include how plasma will affect the propellant through radiation, the deliverance of mechanical energy and heat directly and by driving gas flow. Despite these daunting tasks FLARE has been seen as the most plausible igniter for future application on ETC guns. | [
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10382_3 | Triple coaxial plasma igniter
A coaxial igniter consists of a fully insulated conductor, covered by four strips of aluminum foil. All of this is further insulated in a tube about 1.6 cm in diameter that is perforated with small holes. The idea is to use an electrical flow through the conductor and then exploding the flow into vapor and then breaking it down into plasma. Consequently, the plasma escapes through the constant perforations throughout the insulating tube and initiates the surrounding propellant. A TCPI igniter is fitted in individual propellant cases for each piece of ammunition. However, TCPI is no longer considered a viable method of propellant ignition because it may damage the fins and does not deliver energy as efficiently as a FLARE igniter.
Feasibility
The XM291 is the best existing example of a working electrothermal-chemical gun. It was an alternative technology to the heavier caliber 140 mm gun by using the dual-caliber approach. It uses a breech that is large enough to accept 140 mm ammunition and be mounted with both a 120 mm barrel and a 135 mm or 140 mm barrel. The XM291 also mounts a larger gun tube and a larger ignition chamber than the existing M256 L/44 main gun. Through the application of electrothermal-chemical technology the XM291 has been able to achieve muzzle energy outputs that equate that to a low-level 140 mm gun, while achieving muzzle velocities greater than those of the larger 140 mm gun. Although the XM291 does not mean that ETC technology is viable it does offer an example that it is possible. | [
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10382_4 | ETC is also a more viable option than other alternatives by definition. ETC requires much less energy input from outside sources, like a battery, than a railgun or a coilgun would. Tests have shown that energy output by the propellant is higher than energy input from outside sources on ETC guns. In comparison, a railgun currently cannot achieve a higher muzzle velocity than the amount of energy input. Even at 50% efficiency a rail gun launching a projectile with a kinetic energy of 20 MJ would require an energy input into the rails of 40 MJ, and 50% efficiency has not yet been achieved. To put this into perspective, a rail gun launching at 9 MJ of energy would need roughly 32 MJ worth of energy from capacitors. Current advances in energy storage allow for energy densities as high as 2.5 MJ/dm³, which means that a battery delivering 32 MJ of energy would require a volume of 12.8 dm³ per shot; this is not a viable volume for use in a modern main battle tank, especially one designed to be lighter than existing models. There has even been discussion about eliminating the necessity for an outside electrical source in ETC ignition by initiating the plasma cartridge through a small explosive force.
Furthermore, ETC technology is not only applicable to solid propellants. To increase muzzle velocity even further electrothermal-chemical ignition can work with liquid propellants, although this would require further research into plasma ignition. ETC technology is also compatible with existing projects to reduce the amount of recoil delivered to the vehicle while firing. Understandably, recoil of a gun firing a projectile at 17 MJ or more will increase directly with the increase in muzzle energy in accordance to Newton's third law of motion and successful implementation of recoil reduction mechanisms will be vital to the installation of an ETC powered gun in an existing vehicle design. For example, OTO Melara's new lightweight 120 mm L/45 gun has achieved a recoil force of 25 t by using a longer recoil mechanism (550 mm) and a pepperpot muzzle brake. Reduction in recoil can also be achieved through mass attenuation of the thermal sleeve. The ability of ETC technology to be applied to existing gun designs means that for future gun upgrades there's no longer the necessity to redesign the turret to include a larger breech or caliber gun barrel. | [
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10382_5 | Several countries have already determined that ETC technology is viable for the future and have funded indigenous projects considerably. These include the United States, Germany and the United Kingdom, amongst others. The United States' XM360, which was planned to equip the Future Combat Systems Mounted Combat System light tank and may be the M1 Abrams' next gun upgrade, is reportedly based on the XM291 and may include ETC technology, or portions of ETC technology. Tests of this gun have been performed using "precision ignition" technology, which may refer to ETC ignition.
Notes
Bibliography
External links
Electromagnetic Launch Symposium
http://www.powerlabs.org/electrothermal.htm
Artillery by type
Emerging technologies
Propellants
History of the tank | [
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10384_0 | The Boeing E-3 Sentry is an American airborne early warning and control (AEW&C) aircraft developed by Boeing. E-3s are commonly known as AWACS (Airborne Warning and Control System). Derived from the Boeing 707 airliner, it provides all-weather surveillance, command, control, and communications, and is used by the United States Air Force, NATO, French Air and Space Force, and Royal Saudi Air Force. The E-3 is distinguished by the distinctive rotating radar dome (rotodome) above the fuselage. Production ended in 1992 after 68 aircraft had been built.
In the mid-1960s, the U.S. Air Force (USAF) was seeking an aircraft to replace its piston-engined Lockheed EC-121 Warning Star, which had been in service for over a decade. After issuing preliminary development contracts to three companies, the USAF picked Boeing to construct two airframes to test Westinghouse Electric and Hughes's competing radars. Both radars used pulse-Doppler technology, with Westinghouse's design emerging as the contract winner. Testing on the first production E-3 began in October 1975.
The first USAF E-3 was delivered in March 1977, and during the next seven years, a total of 34 aircraft were manufactured. E-3s were also purchased by NATO (18), the United Kingdom (7), France (4) and Saudi Arabia (5).
In 1991, when the last aircraft had been delivered, E-3s participated in the Persian Gulf War, playing a crucial role of directing coalition aircraft against Iraqi forces. The aircraft's capabilities have been maintained and enhanced through numerous upgrades. In 1996, Westinghouse Electric's Defense & Electronic Systems division was acquired by Northrop Corporation, before being renamed Northrop Grumman Mission Systems, which currently supports the E-3's radar.
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] |
10384_1 | Development
Background
In 1963, the USAF asked for proposals for an Airborne Warning and Control System (AWACS) to replace its EC-121 Warning Stars, which had served in the airborne early warning role for over a decade. The new aircraft would take advantage of improvements in radar technology and computer-aided radar data analysis and data reduction. These developments allowed airborne radars to "look down", i.e. to detect the movement of low-flying aircraft, and discriminate, even over land, target aircraft's movements; previously this had been impossible due to the inability to discriminate an aircraft's track from ground clutter. Contracts were issued to Boeing, Douglas, and Lockheed, the latter being eliminated in July 1966. In 1967, a parallel program was put into place to develop the radar, with Westinghouse Electric Corporation and Hughes Aircraft being asked to compete in producing the radar system. In 1968, it was referred to as Overland Radar Technology (ORT) during development tests on the modified EC-121Q. The Westinghouse radar antenna was going to be used by whichever company won the radar competition since Westinghouse had pioneered the design of high-power radio frequency (RF) phase-shifters, which are used to both focus the RF into a pencil beam and scan electronically for altitude determination.
Boeing initially proposed a purpose-built aircraft, but tests indicated it would not outperform the already-operational 707, so the latter was chosen instead. To increase endurance, this design was to be powered by eight General Electric TF34s. It would carry its radar in a rotating dome mounted at the top of a forward-swept tail, above the fuselage. Boeing was selected ahead of McDonnell Douglas's DC-8-based proposal in July 1970. Initial orders were placed for two aircraft, designated EC-137D, as test beds to evaluate the two competing radars. As the test beds did not need the same 14-hour endurance demanded of the production aircraft, the EC-137s retained the Pratt & Whitney JT3D commercial engines, and a later reduction in the endurance requirement led to retention of the JT3D engines in production. | [
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10384_2 | The first EC-137 made its maiden flight on 9 February 1972, with the fly-off between the two radars taking place from March to July of that year. Favorable test results led to the selection of Westinghouse's radar for the production aircraft. Hughes' radar was initially thought to be a certain winner due to its related development of the APG-63 radar for the new F-15 Eagle. The Westinghouse radar used a pipelined fast Fourier transform (FFT) to digitally resolve 128 Doppler frequencies, while Hughes's radars used analog filters based on the design for the F-15. Westinghouse's engineering team won this competition by using a programmable 18-bit computer whose software could be modified before each mission. This computer was the AN/AYK-8 design from the B-57G program, and designated AYK-8-EP1 for its much expanded memory. This radar also multiplexed a beyond-the-horizon (BTH) pulse mode that could complement the pulse-Doppler radar mode. This proved to be beneficial especially when the BTH mode is used to detect ships at sea when the radar beam is directed below the horizon.
Full-scale development
Approval was given on 26 January 1973 for the full-scale development of the AWACS system. To allow further development of the aircraft's systems, orders were placed for three preproduction aircraft, the first of which performed its maiden flight in February 1975. IBM and Hazeltine were selected to develop the mission computer and display system. The IBM computer was designated 4PI, and the software was written in JOVIAL. A Semi-Automatic Ground Environment (SAGE) or back-up interceptor control (BUIC) operator would immediately be at home with the track displays and tabular displays, but differences in symbology would create compatibility problems in tactical ground radar systems in Iceland, mainland Europe, and South Korea over Link-11 (TADIL-A). In 1977, Iran placed an order for ten E-3s, however this order was cancelled following the Iranian Revolution.
Engineering, test and evaluation began on the first E-3 Sentry in October 1975. Between 1977 and 1992, a total of 68 E-3s were built. | [
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10384_3 | Future status
Because the Boeing 707 is no longer in production, the E-3 mission package has been fitted into the Boeing E-767 for the Japan Air Self Defense Forces. The E-10 MC2A was intended to replace USAF E-3s—along with the RC-135 and the E-8 Joint STARS, but the program was canceled by the Department of Defense.
NATO intends to extend the operational status of its AWACS until 2035 when it is due to be replaced by the Alliance Future Surveillance and Control (AFSC) program. The Royal Air Force (RAF) chose to limit investment in its E-3D fleet in the early 2000s, diverting Sentry upgrade funds to a replacement program. On 22 March 2019, the UK Defence Secretary announced a $1.98 billion contract to purchase five E-7 Wedgetails.
Design
Overview
The E-3 Sentry's airframe is a modified Boeing 707-320B Advanced model. Modifications include a rotating radar dome (rotodome), uprated hydraulics from 241 to 345 bar (3500–5000 PSI) to drive the rotodome, single-point ground refueling, air refueling, and a bail-out tunnel or chute. A second bail-out chute was deleted to cut mounting costs.
USAF and NATO E-3s have an unrefueled range of or 8 hours of flying. The newer E-3 versions bought by France, Saudi Arabia, and the UK are equipped with newer CFM56-2 turbofan engines, and these can fly for about 11 hours or more than . The Sentry's range and on-station time can be increased through air-to-air refueling and the crews can work in shifts by the use of an on-board crew rest and meals area. The aircraft are equipped with one toilet in the rear, and a urinal behind the cockpit. Saudi E-3s were delivered with an additional toilet in the rear. | [
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10384_4 | When deployed, the E-3 monitors an assigned area of the battlefield and provides information for commanders of air operations to gain and maintain control of the battle; while as an air defense asset, E-3s can detect, identify, and track airborne enemy forces far from the boundaries of the U.S. or NATO countries and can direct interceptor aircraft to these targets. In support of air-to-ground operations, the E-3 can provide direct information needed for interdiction, reconnaissance, airlift, and close-air support for friendly ground forces.
Avionics
The unpressurized rotodome is in diameter, thick at the center, and is held above the fuselage by 2 struts. It is tilted down at the front to reduce its aerodynamic drag, which lessens its detrimental effect on take-offs and endurance. This tilt is corrected electronically by both the radar and secondary surveillance radar antenna phase shifters. The rotodome uses bleed air, outside cooling doors, and fluorocarbon-based cold plate cooling to maintain the electronic and mechanical equipment temperatures. The hydraulically rotated antenna system permits the and AN/APY-2 passive electronically scanned array radar system to provide surveillance from the Earth's surface up into the stratosphere, over land or water.
Other major subsystems in the E-3 Sentry are navigation, communications, and computers. 14 consoles display computer-processed data in graphic and tabular format on screens. Its operators perform surveillance, identification, weapons control, battle management and communications functions. Data may be forwarded in real-time to any major command and control center in rear areas or aboard ships. In times of crisis, data may also be forwarded to the National Command Authority in the U.S. via RC-135 or aircraft carrier task forces. | [
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] |
10384_5 | Electrical generators mounted in each of the E-3's four engines provide 1 megawatt of electrical power required by the aircraft's radars and electronics. Its pulse-Doppler radar has a range of more than 250 mi (400 km) for low-flying targets at its operating altitude, and the pulse (BTH) radar has a range of approximately 400 mi (650 km) for aircraft flying at medium to high altitudes. The radar, combined with a secondary surveillance radar (SSR) and electronic support measures (ESM), provides a look down capability, to detect, identify, and track low-flying aircraft, while eliminating ground clutter returns.
Upgrades
Between 1987 and 2001, USAF E-3s were upgraded under the "Block 30/35 Modification Program". Enhancements included:
The installation of ESM and an electronic surveillance capability, for both active and passive means of detection.
Installation of the Joint Tactical Information Distribution System (JTIDS), which provides rapid and secure communication for transmitting information, including target positions and identification data, to other friendly platforms.
Global Positioning System (GPS) capability was added.
Onboard computers were overhauled to accommodate JTIDS, Link-16, the new ESM systems and to allow for future enhancements.
RSIP
The Radar System Improvement Program (RSIP) was a joint US/NATO development program. RSIP enhances the operational capability of the E-3 radars' electronic countermeasures, and improves the system's reliability, maintainability, and availability. Essentially, this program replaced the older transistor-transistor logic (TTL) and emitter-coupled logic (MECL) electronic components, long-since out of production, with off-the-shelf computers that utilised a High-level programming language instead of assembly language. Significant improvement came from adding pulse compression to the pulse-Doppler mode. These hardware and software modifications improve the E-3 radars' performance, providing enhanced detection with an emphasis towards low radar cross-section (RCS) targets. | [
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10384_6 | The RAF had also joined the USAF in adding RSIP to upgrade the E-3's radars. The retrofitting of the E-3 squadrons was completed in December 2000. Along with the RSIP upgrade was installation of the Global Positioning System/Inertial Navigation Systems which improved positioning accuracy. In 2002, Boeing was awarded a contract to add RSIP to the small French AWACS squadron. Installation was completed in 2006. Saudi Arabia began RSIP upgrades in 2013; the first aircraft being upgraded by Boeing in Seattle with the four remaining aircraft upgraded in Riyadh between 2014 and 2016.
NATO Mid Term Program
Between 2000 and 2008 NATO upgraded its E-3s to Mid Term Program (MTP) standard. This involved technical upgrades and a total multi-sensor-systems integration
DRAGON
In 2009, the USAF, in cooperation with NATO, entered into a major flight deck avionics modernization program in order to maintain compliance with worldwide airspace mandates. The program, called DRAGON (for DMS Replacement of Avionics for Global Operation and Navigation), was awarded to Boeing and Rockwell Collins in 2010. Drawing on their Flight2 flight management system (FMS), almost all the avionics were replaced with more modern digital equipment from Rockwell Collins. Main upgrades include a Digital Audio Distribution System, Mode-5/ADS-B transponder, Inmarsat and VDL datalinks, and a terrain awareness and warning system (TAWS). The centerpiece flight deck hardware consists of five 6x8 color graphics displays and two color CDUs. DRAGON laid the foundation for subsequent upgrades including GPS M-Code, Iridium ATC, and Autopilot. USAF DRAGON Production began in 2018. | [
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10384_7 | USAF Block 40/45
In 2014 the USAF began upgrading block 30/35 E-3B/Cs into block 40/45 E-3Gs. This upgrade replaces the main flight computer with a Red Hat Linux-based system, as well as replacing the DOS 2.0-like operating system with a Windows 95-like system on the operator workstations. In 2016, a three-week long cybersecurity vulnerability test revealed that the 40/45 block and its supporting ground equipment were vulnerable to cyber threats, and were thus deemed "not survivable." This caused a delay of approximately two years. Twenty-four E-3s are projected to complete this upgrade to 40/45 by the end of fiscal year 2020, while seven aircraft will be retired to save upgrade costs and harvest out-of-production components.
NATO Final Lifetime Extension Program
NATO intends to extend the operational status of its AWACS until 2035 by significantly upgrading fourteen aircraft in the Final Lifetime Extension Program (FLEP) between 2019 and 2026. Upgrades include the expansion of data capacity, expansion of bandwidth for satellite communications, new encryption equipment, new HAVE QUICK radios, upgraded mission computing software and new operator consoles. The supporting ground systems (mission training center and mission planning and evaluation system) will also be upgraded to the latest standard. NATO Airborne Early Warning & Control Program Management Agency (NAPMA) is the preparing and executing authority for the FLEP. FLEP will be combined with the standard planned higher echelon technical maintenance.
Operational history
In March 1977 the 552nd Airborne Warning and Control Wing received the first E-3 aircraft at Tinker AFB, Oklahoma. The 34th and last USAF Sentry was delivered in June 1984. The USAF has a total of thirty-one E-3s in active service. Twenty-seven are stationed at Tinker AFB and belong to the Air Combat Command (ACC). Four are assigned to the Pacific Air Forces (PACAF) and stationed at Kadena AB, Okinawa and Elmendorf AFB, Alaska. One aircraft (TS-3) was assigned to Boeing for testing and development (retired/scrapped June 2012). | [
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10384_8 | E-3 Sentry aircraft were among the first to deploy during Operation Desert Shield, where they established a radar screen to monitor Iraqi forces. During Operation Desert Storm, E-3s flew 379 missions and logged 5,052 hours of on-station time. The data collection capability of the E-3 radar and computer subsystems allowed an entire air war to be recorded for the first time. In addition to providing senior leadership with time-critical information on the actions of enemy forces, E-3 controllers assisted in 38 of the 41 air-to-air kills recorded during the conflict.
NATO, UK, French and USAF AWACS played an important role in the air campaign against Serbia and Montenegro in the former republic of FR Yugoslavia. From March to June 1999 the aircraft were deployed in the NATO bombing of Yugoslavia (operation Allied Force) directing allied strike and air defence aircraft to and from their targets. Over 1,000 aircraft operating from bases in Germany and Italy took part in the air campaign which was intended to destroy Yugoslav air defenses and high-value targets such as the bridges across the Danube river, factories, power stations, telecommunications facilities, and military installations.
On 18 November 2015, an E-3G was deployed to the Middle East to begin flying combat missions in support of Operation Inherent Resolve against ISIL, marking the first combat deployment of the upgraded Block 40/45 aircraft.
France and United Kingdom
In February 1987 the UK and France ordered E-3 aircraft in a joint project which saw deliveries start in 1991. The British requirement arose due to the cancellation of the BAE Nimrod AEW3 project. While France operates its E-3F aircraft independently of NATO, UK E-3Ds formed the E-3D Component of the NATO Airborne Early Warning and Control Force (NAEWCF), receiving much of their tasking directly from NATO. However, RAF E-3Ds remain UK manned and capable of independent, national tasking. This has been done on numerous occasions, notably when E-3Ds were committed to operations over Afghanistan in 2001 and Iraq in 2003. | [
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10384_9 | The UK fleet has slowly been reduced from 7 since 2011. In 2009, the UK effectively limited the service life of the E-3D fleet by de-funding the Project Eagle upgrade which would have seen it upgraded in line with the USAF Block 40/45 standard. AirForces Monthly reported that by December 2020, just 2 aircraft were available for operations at any one time. The Strategic Defence and Security Review 2015 had announced the intention to retain the E-3D fleet until 2035, however in March 2019, the Ministry of Defence announced that the E-3Ds would be replaced by five E-7 Wedgetails from 2023. The £1.51 billion contract was awarded to Boeing without a competitive procurement process, a decision criticised by both competitors of Boeing and the UK's Defence Select Committee. The 2021 Integrated Defence Review confirmed a reduced order of three aircraft. On 27 January 2015, the RAF deployed an E-3D Sentry to Cyprus in support of U.S.-led coalition airstrikes against Islamic State militants in Iraq and Syria. The last intended operational flight by an RAF E3 Sentry was supposed to be in July 2021 with the Sentry retired from service, however RAF ZH101 and ZH106 have both flown patrols over Poland / Eastern Europe during Russia's incursions into Ukraine during late February / March 2022.
France operates four aircraft, all fitted with the newer CFM56-2 engines.
NATO
NATO acquired 18 E-3As and support equipment, with the first aircraft delivered in January 1982. The aircraft are registered in Luxembourg. The eighteen E-3s were operated by Number 1, 2 and 3 Squadrons of NATO's E-3 Component, based at NATO Air Base Geilenkirchen.
NATO E-3s participated in Operation Eagle Assist after the September 11 attacks on the World Trade Center towers and the Pentagon. NATO and RAF E-3s participated in the military intervention in Libya. | [
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10384_10 | Presently, 16 NATO E-3As are in the inventory, since one E-3 was lost in a crash and one was retired from service in 2015. The latter was due for its six-year cycle Depot Level Maintenance (DLM) inspection which would have been very costly. The "449 Retirement Project" resulted in reclamation of critical parts with a value of upwards of $40 million which will be used to support the 16 active aircraft. Some of the parts to be removed are no longer on the market or have become very expensive.
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] |
10384_11 | Variants
EC-137D
2 prototype AWACS aircraft with JT3D engines, 1 fitted with a Westinghouse Electric radar and 1 with a Hughes Aircraft Company radar. Both converted to E-3A standard with TF33 engines.
E-3A
Production aircraft with TF33 engines and AN/APY-1 radar, 24 built for USAF (later converted to E-3B standard), total of 34 ordered but the last 9 completed as E-3C. One additional aircraft retained by Boeing for testing, 18 built for NATO with TF33 engines and 5 for Saudi Arabia with CFM56 engines.
KE-3A
These are not AWACS aircraft but CFM56 powered tankers based on the E-3 design. 8 were sold to Saudi Arabia.
E-3B
USAF Block 30 modification. E-3As with improvements, 24 conversions.
E-3C
USAF Block 35 modification. Production aircraft with AN/APY-2 radar, additional electronic consoles and system improvements, ten built.
JE-3C
One E-3A aircraft used by Boeing for trials later redesignated E-3C.
E-3D
Production aircraft for the RAF to E-3C standard with CFM56 engines and British modifications designated Sentry AEW.1, 7 built. Modifications included the addition of a refuelling probe next to the existing boom AAR recipticle, CFM-56 engines, wingtip ESM pods, an enhanced Maritime Surveillance Capability (MSC) offering Maritime Scan-Scan Processing (MSSP), JTIDS and Havequick 2 radios.
E-3F
Production aircraft for the French Air and Space Force to E-3C standard with CFM56 engines and French modifications, 4 built.
E-3G
USAF Block 40/45 modification. Includes hardware and software upgrades to improve communications, computer processing power, threat tracking, and others, and automates some previously manual functions. Initial operating capability (IOC) declared in July 2015.
Operators
The Chilean Air Force purchased three second-hand E-3D Sentry aicraft from the Royal Air Force.
The French Air and Space Force purchased four E-3F aircraft.
Escadron de détection et de contrôle aéroportés 36 Berry (36th Airborne Detection and Control Squadron "Berry") based at Avord Air Base. | [
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] |
10384_12 | 18 E-3 AWACS were purchased – 1 was written off in Greece, 3 were retired from service. Mainly responsible for monitoring European NATO airspace, they have also been deployed outside the area in support of NATO commitments. The 20 multinational crews are provided by 15 of the 28 NATO member states.
NATO Airborne Early Warning and Control Force – E-3A Component. Based at Geilenkirchen (Germany), with forward operating bases at Konya (Turkey), Preveza/Aktion (Greece) and Trapani/Birgi (Italy) and a forward operating location at Ørland (Norway).
Aircrew Training Squadron
Flying Squadron 1
Flying Squadron 2
Flying Squadron 3 – disbanded in 2015
The Royal Saudi Air Force purchased five E-3A aircraft in 1983. In 2004, modifications began to convert KE-3A tankers into RE-3 electronic intelligence gathering aircraft.
RSAF No. 6 Wing (Prince Sultan Air Base – Al Kharj)
السرب الثامن عشر (al-Sarab al-Ththamin Eshr – No. 18 Squadron)
No. 19 Squadron – RE-3A/B (as well as Beechcraft 350ER-ISR)
No. 23 Squadron – KE-3A
The Royal Air Force purchased seven E-3Ds by October 1987, designated Sentry AEW.1 in British service. As of December 2020, only three remained in service after one was withdrawn from service in 2009 to be used as spares, two were withdrawn in March 2019 and a further one withdrawn in January 2020. The fleet had been given an out of service date (OSD) of December 2022. They form the E-3D Component of the NATO Airborne Early Warning and Control Force. However, that date was accelerated pursuant to the 2021 defence review and the aircraft made its final flight in U.K. service in August 2021.
RAF Waddington, Lincolnshire, England
No. 8 Squadron (1991–2021)
No. 23 Squadron (1996–2009)
No. 54 Squadron (Operational Conversion Unit 2005–?)
No. 56 Squadron (Operational Evaluation Unit 2008–?) | [
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10384_13 | The United States Air Force has 31 operational E-3s as of December 2019
Tactical Air Command 1976–1992
Air Combat Command 1992–present
552d Air Control Wing – Tinker Air Force Base, Oklahoma
960th Airborne Air Control Squadron 2001–present (NAS Keflavik, Iceland 1979–1992)
963d Airborne Air Control Squadron 1976–present
964th Airborne Air Control Squadron 1977–present
965th Airborne Air Control Squadron 1978–1979, 1984–present
966th Airborne Air Control Squadron 1976–present
380th Air Expeditionary Wing – Al Dhafra Air Base, United Arab Emirates
968th Expeditionary Airborne Air Control Squadron 2013–present (Thumrait Air Base, Oman 2002–2003)
Air Force Reserve Command
513th Air Control Group (Associate) – Tinker AFB, Oklahoma
970th Airborne Air Control Squadron 1996–present (Personnel only, aircraft loaned by co-located 552nd ACW as needed)
413th Flight Test Group – Robins AFB, Georgia
10th Flight Test Squadron – Tinker AFB, Oklahoma 1994–present
Pacific Air Forces
3d Wing – Elmendorf AFB, Alaska
962d Airborne Air Control Squadron 1986–present
18th Wing – Kadena AB, Japan
961st Airborne Air Control Squadron 1979–present
Incidents and accidents
E-3s have been involved in three hull-loss accidents, and one radar antenna was destroyed during RSIP development (see photo under Avionics).
On 22 September 1995, a U.S. Air Force E-3 Sentry (callsign Yukla 27, serial number 77-0354), crashed shortly after takeoff from Elmendorf AFB, Alaska. The plane lost power to both left side engines after ingesting several Canada geese during takeoff. The aircraft went down about northeast of the runway, killing all 24 crew members on board.
On 14 July 1996, a NATO E-3 Sentry (tail number LX-N90457) overran the runway and crashed into a sea wall at Préveza-Aktion Airport in Greece when the pilot attempted to abort takeoff after mistakenly believing that the aircraft had suffered a bird strike. The aircraft overran the runway and struck a sea wall, where it came to a halt. There were no injuries and the aircraft was written off. Investigators could find no evidence that a bird strike and ingestion had occurred. | [
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] |
10384_14 | On 28 August 2009, a U.S. Air Force E-3C Sentry (serial number 83-0008) participating in a Red Flag exercise at Nellis Air Force Base, Nevada experienced a nose gear collapse on landing, resulting in a fire and damaging the aircraft beyond repair. All 32 crew members evacuated safely.
Specifications (USAF/NATO)
See also
References
Notes
Citations
Bibliography
External links
Royal Air Force E-3 Sentry information
NATO AWACS-Spotter Geilenkirchen website
Airborne Early Warning Association website
Aircraft first flown in 1972
AWACS aircraft
E-03 Sentry
Boeing 707
E-03 Sentry
Quadjets
1970s United States military reconnaissance aircraft | [
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10385_0 | The Northrop Grumman E-8 Joint Surveillance Target Attack Radar System (Joint STARS) is a United States Air Force airborne ground surveillance, battle management and command and control aircraft. It tracks ground vehicles and some aircraft, collects imagery, and relays tactical pictures to ground and air theater commanders. The aircraft is operated by both active duty U.S. Air Force and Air National Guard units and also carries specially trained U.S. Army personnel as additional flight crew.
Development
Joint STARS evolved from separate United States Army and Air Force programs to develop technology to detect, locate and attack enemy armor at ranges beyond the forward area of troops. In 1982, the programs were merged and the U.S. Air Force became the lead agent. The concept and sensor technology for the E-8 was developed and tested on the Tacit Blue experimental aircraft. The prime contract was awarded to Grumman Aerospace Corporation in September 1985 for two E-8A development systems.
Upgrades
In late 2005, Northrop Grumman was awarded a contract for upgrading engines and other systems. Pratt & Whitney, in a joint venture with Seven Q Seven (SQS), will produce and deliver JT8D-219 engines for the E-8s. Their greater efficiency will allow the Joint STARS to spend more time on station, take off from a wider range of runways, climb faster, fly higher all with a much reduced cost per flying hour.
In December 2008, an E-8C test aircraft took its first flight with the new engines. In 2009, the company began engine replacement and additional upgrade efforts. However, the re-engining funding was temporarily halted in 2009 as the Air Force began to consider other options for performing the JSTARS mission.
Design
The E-8C is an aircraft modified from the Boeing 707-300 series commercial airliner. The E-8 carries specialized radar, communications, operations and control subsystems. The most prominent external feature is the 40 ft (12 m) canoe-shaped radome under the forward fuselage that houses the 24 ft (7.3 m) APY-7 active electronically scanned array side looking airborne radar antenna. | [
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10385_1 | The E-8C can respond quickly and effectively to support worldwide military contingency operations. It is a jam-resistant system capable of operating while experiencing heavy electronic countermeasures. The E-8C can fly a mission profile for 9 hours without refueling. Its range and on-station time can be substantially increased through in-flight refueling.
Radar and systems
The AN/APY-7 radar can operate in wide area surveillance, ground moving target indicator (GMTI), fixed target indicator (FTI) target classification, and synthetic aperture radar (SAR) modes.
To pick up moving targets, the Doppler radar looks at the Doppler frequency shift of the returned signal. It can look from a long-range, which the military refers to as a high standoff capability. The antenna can be tilted to either side of the aircraft for a 120-degree field of view covering nearly 19,305 square miles (50,000 km2) and can simultaneously track 600 targets at more than 152 miles (250 km). The GMTI modes cannot pick up objects that are too small, insufficiently dense, or stationary. Data processing allows the APY-7 to differentiate between armored vehicles (tracked tanks) and trucks, allowing targeting personnel to better select the appropriate ordnance for various targets.
The system's SAR modes can produce images of stationary objects. Objects with many angles (for example, the interior of a pick-up bed) will give a much better radar signature, or specular return. In addition to being able to detect, locate and track large numbers of ground vehicles, the radar has a limited capability to detect helicopters, rotating antennas and low, slow-moving fixed-wing aircraft.
The radar and computer subsystems on the E-8C can gather and display broad and detailed battlefield information. Data is collected as events occur. This includes position and tracking information on enemy and friendly ground forces. The information is relayed in near-real time to the US Army's common ground stations via the secure jam-resistant surveillance and control data link (SCDL) and to other ground C4I nodes beyond line-of-sight via ultra high-frequency satellite communications. | [
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10385_2 | Other major E-8C prime mission equipment are the communications/datalink (COMM/DLX) and operations and control (O&C)subsystems. Eighteen operator workstations display computer-processed data in graphic and tabular format on video screens. Operators and technicians perform battle management, surveillance, weapons, intelligence, communications and maintenance functions.
Northrop Grumman has tested the installation of a MS-177 camera on an E-8C to provide real time visual target confirmation.
Battle management
In missions from peacekeeping operations to major theater war, the E-8C can provide targeting data and intelligence for attack aviation, naval surface fire, field artillery and friendly maneuver forces. The information helps air and land commanders to control the battlespace.
The E-8's ground-moving radar can tell approximate number of vehicles, location, speed, and direction of travel. It cannot identify exactly what type of vehicle a target is, tell what equipment it has, or discern whether it is friendly, hostile, or a bystander, so commanders often crosscheck the JSTARS data against other sources. In the Army, JSTARS data is analyzed in and disseminated from a Ground Station Module (GSM).
Operational history
The two E-8A development aircraft were deployed in 1991 to participate in Operation Desert Storm under the direction of USAF Colonel Harry H. Heimple, Program Director, even though they were still in development. The joint program accurately tracked mobile Iraqi forces, including tanks and Scud missiles. Crews flew developmental aircraft on 49 combat sorties, accumulating more than 500 combat hours and a 100% mission effectiveness rate.
These Joint STARS developmental aircraft also participated in Operation Joint Endeavor, a NATO peacekeeping mission, in December 1995. While flying in friendly air space, the test-bed E-8A and pre-production E-8C aircraft monitored ground movements to confirm compliance with the Dayton Peace Accords agreements. Crews flew 95 consecutive operational sorties and more than 1,000 flight hours with a 98% mission effectiveness rate. | [
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10385_3 | The 93d Air Control Wing, which activated 29 January 1996, accepted its first aircraft on 11 June 1996, and deployed in support of Operation Joint Endeavor in October. The provisional 93d Air Expeditionary Group monitored treaty compliance while NATO rotated troops through Bosnia and Herzegovina. The first production E-8C and a pre-production E-8C flew 36 operational sorties and more than 470 flight hours with a 100% effectiveness rate. The wing declared initial operational capability 18 December 1997 after receiving the second production aircraft. Operation Allied Force saw Joint STARS in action again from February to June 1999 accumulating more than 1,000 flight hours and a 94.5% mission-effectiveness rate in support of the U.S.-lead Kosovo War.
On 1 October 2002, the 93d Air Control Wing (93 ACW) was "blended" with the 116th Bomb Wing in a ceremony at Robins Air Force Base, Georgia. The 116 BW was an Air National Guard wing equipped with the B-1B Lancer bomber at Robins AFB. As a result of a USAF reorganization of the B-1B force, all B-1Bs were assigned to active duty wings, resulting in the 116 BW lacking a current mission. The newly created wing was designated as the 116th Air Control Wing (116 ACW). The 93 ACW was inactivated the same day. The 116 ACW constituted the first fully blended wing of active duty and Air National Guard airmen.
The wing took delivery of the 17th and final E-8C on 23 March 2005. The E-8C Joint STARS routinely supports various taskings of the Combined Force Command Korea during the North Korean winter exercise cycle and for the United Nations enforcing resolutions on Iraq. The twelfth production aircraft, outfitted with an upgraded operations and control subsystem, was delivered to the USAF on 5 November 2001.
In March 2009, a Joint STARS aircraft was damaged beyond economical repair when a test plug was left on a fuel tank vent, subsequently causing the fuel tank to rupture during in-flight refueling. There were no casualties but the aircraft sustained $25 million in damage. | [
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10385_4 | In September 2009, Loren B. Thompson of the Lexington Institute raised the question of why most of the Joint STARS fleet was sitting idle instead of being used to track insurgents in Afghanistan. Thompson states that the Joint STARS' radar has an inherent capacity to find what the Army calls 'dismounted' targets—insurgents walking around or placing roadside bombs. Thompson's neutrality has been questioned by some since Lexington Institute has been heavily funded by defense contractors, including Northrop Grumman.
Recent trials of Joint STARS in Afghanistan are destined to develop tactics, techniques and procedures in tracking dismounted, moving groups of Taliban.
In January 2011, Northrop Grumman's E-8C Joint Surveillance Target Attack Radar System (Joint STARS) test bed aircraft completed the second of two deployments to Naval Air Station Point Mugu, California, in support of the U.S. Navy Joint Surface Warfare Joint Capability Technology Demonstration to test its Network-Enabled Weapon (NEW) architecture. The Joint STARS aircraft executed three Operational Utility Assessment flights and demonstrated its ability to guide anti-ship weapons against surface combatants at a variety of standoff distances in the NEW architecture.
From 2001 to January 2011 the Joint STARS fleet flew more than 63,000 hours in 5,200 combat missions in support of Operations Iraqi Freedom, Enduring Freedom and New Dawn.
On 1 October 2011, the "blended" wing construct of the 116th Air Control Wing (116 ACW), combining Air National Guard and Regular Air Force personnel in a single unit was discontinued. On this date, the 461st Air Control Wing (461 ACW) was established at Robins AFB as the Air Force's sole active duty E-8 Joint STARS wing while the 116 ACW reverted to a traditional Air National Guard wing within the Georgia Air National Guard. Both units share the same E-8 aircraft and will often fly with mixed crews, but now function as separate units.
On 1 October 2019, JSTARS ended its continuous presence in the United States Central Command (USCENTCOM) areas of responsibility. The 18–year deployment was the second-longest deployment in U.S. Air Force history. In that time, the crews and aircraft flew 10,938 sorties, and 114,426.6 combat hours. | [
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10385_5 | On 11 February 2022, the first of four JSTARS out of the remaining 16 operational JSTARS was retired as detailed in the Fiscal Year 2022 National Defence Authorisation Act (NDAA). The plane (serial number 92-3289/GA) which was the first to arrive at Robins AFB in 1996 has now been transferred to the 309th Aerospace Maintenance and Regeneration Group at Davis–Monthan Air Force Base.
In late 2021 to early 2022, E-8C JSTARS aircraft deployed to Europe during the 2021-2022 Russo-Ukrainian crisis. Ironically, 30 years after entering service, it was performing the type of mission it had originally been intended to: monitoring Russian military activity in Eastern Europe, which it did until the outbreak of the Russo-Ukrainian War in late February 2022.
Future
The Air Force began an analysis of alternatives (AOA) in March 2010 for its next generation ground GMTI radar aircraft fleet. The study was completed in March 2012 and recommended buying a new business jet-based ISR aircraft, such as a version of the P-8 Poseidon, and the RQ-4B Global Hawk Block 40. The Air Force says Joint STARS is in a phase of capability improvements and is expected to remain in operation through 2030.
On 23 January 2014, the USAF revealed a plan for the acquisition of a new business jet-class replacement for the E-8C Joint STARS. The program is called Joint STARS Recap and plans for the aircraft to reach initial operating capability (IOC) by 2022. The airframe must be more efficient, and separate contracts will be awarded for developing the aircraft, airborne sensor, battle management command and control (BMC2) system, and communications subsystem.
On 8 April 2014, the Air Force held an industry day for companies interested in competing for JSTARS Recap; attendees included Boeing, Bombardier Aerospace, and Gulfstream Aerospace. Air Force procurement documents called for a replacement for the Boeing 707-based E-8C as a "business jet class" airframe that is "significantly smaller and more efficient." Indicative specification were for an aircraft with a 10-13 person crew with a radar array and capable of flying at 38,000 ft for eight hours. | [
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] |
10385_6 | In August 2015, the Air Force issued contracts to Boeing, Lockheed Martin, and Northrop Grumman for a one-year pre-engineering and manufacturing development effort to mature and test competing designs ahead of a downselect in late 2017.
During the fiscal 2019 budget rollout briefing it was announced that the Air Force will not move forward with an E-8C replacement aircraft. Funding for the JSTARS recapitalization program was instead be diverted to pay for development of an advanced battle management system.
Variants
E-8A
Original platform configuration.
TE-8A
Single aircraft with mission equipment removed, used for flight crew training.
YE-8B
Single aircraft, was to be a U.S. Navy E-6 but transferred to the U.S. Air Force as a development aircraft before it was decided to convert second-hand Boeing 707s (1 from a Boeing CC-137) for the JSTARS role.
E-8C
Production Joint STARS platform configuration converted from second-hand Boeing 707s (1 from a CC-137).
Operators
United States Air Force - 17 in service as of Jul. 2018
Air Combat Command - 1991–present
461st Air Control Wing - Robins Air Force Base, Georgia
12th Airborne Command and Control Squadron
16th Airborne Command and Control Squadron
Air National Guard - 2006–present
116th Air Control Wing - Robins Air Force Base, Georgia
128th Airborne Command and Control Squadron
Specifications (E-8C)
See also
References
Citations
Bibliography
External links
Northrop Grumman Joint STARS System Information
Northrop Grumman Joint STARS Radar Information
Boeing Integrated Defense Systems
Northrop Grumman ISR overview
Joint STARS Re-engine Program Info
E-08 Joint STARS
Radar networks
E-08 Joint Stars
Quadjets
E-8 Joint Stars
Aircraft first flown in 1991
E-08 Joint STARS
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10388_0 | Eric Cheney (5 January 1924 – 30 December 2001) was an English motorcycle racer, designer and independent constructor. He was known as one of the best motorcycle frame designers of his era, concentrating mainly in the off-road competition aftermarket.
Early life
Cheney attended a school in Winchester based on Lancasterian principles before joining the Royal Navy at the age of 18, where he served on wartime Arctic convoys and in motor torpedo boats and gained experience of engineering and working on high performance engines. Cheney also worked on the development of remote controlled submarines for the Royal Navy.
Career
After World War II, Cheney joined the motorcycle dealers Archers of Aldershot as a mechanic. Cheney began racing motocross and became one Britain's best riders, along with his travelling companion Les Archer, who went on to become European champion. He had ten successful years on the Continental circuit but a prolonged illness due to an infection contracted while racing in Algeria ended his riding career. He moved into bike preparation and designs for motorcycle chassis and suspension systems.
Cheney had no formal training as a motorcycle designer yet was able to create original and high-performance motorcycle chassis designs working in a simple workshop that was essentially a domestic garage. His approach has been described as "like a medieval engineer" as in an age of computer-aided design and significant resources for research and development teams, he worked entirely by intuition. Eric relied on his long personal experience of international off-road competition riding and would prepare his initial designs for a new motorcycle frame in chalk on the wall of his workshop. Experimenting with different lines until he was satisfied, Cheney would then form the steel tubing using his chalk drawings as a guide. Only when he had built a working prototype motorcycle would he start work on a final jig for mass production. He was once quoted as saying "I know when it's right and it screams at me when it's wrong." | [
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10388_1 | In the late 1960s, the British motorcycle industry was unable to support a national team to compete in the International Six Days Trial so, Cheney hand built a limited number of ISDT Cheney-Triumphs using his own design of twin down-tube frame with a specially tuned Triumph 5TA engine. Fitted with tapered conical hubs, special motocross forks and large alloy fuel tanks, a Cheney Triumph was first used in the 1968 British Trophy Team. In 1970 and 1971 three 504cc Cheney Triumphs were used by the British team in the ISDT, in which Cheney won a manufacturer's prize. Replicas were built, but production was short-lived due to a shortage of engines.
Cheney's most noted successes were in the Grand Prix road racing championships, with Phil Read using his chassis in tandem with a Yamaha engine to win the 1971 250cc world championship. His designs were the last British ones to win a Grand Prix. He never worked for any of the major manufacturers but maintained a productive relationship with BSA in its heyday. After the demise of BSA in 1972, Cheney joined with former BSA factory rider John Banks to develop and campaign a highly successful BSA powered motocross bike.
Some of Cheney's motorcycle designs are now famous in their own right, such as the competition BSA Gold Stars of Jerry Scott and Keith Hickman and the John Banks replica which used a BSA B50 engine specially tuned by Cheney. He also built some racing frames for Suzuki Grand Prix motorcycles in 1968 and, it has been suggested that Suzuki engineers incorporated features of Cheney's designs, such as magnesium hubs and lower fork legs into production road going motorcycles. Cheney's company was originally known as Eric Cheney Designs, then changed to Inter-Moto, now known as Cheney Racing.
Steve McQueen
American actor Steve McQueen, an experienced off-road rider who represented the United States in the ISDT bought a number of Cheney's motorcycles at full price because he considered them better than other makes.
References
External links
Cheney Racing web site
British automobile designers
1924 births
2001 deaths
British motorcycle designers
British motocross riders
People from Hampshire
Royal Navy personnel of World War II | [
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10390_0 | Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships. More precisely, it is "the quantitative analysis of actual economic phenomena based on the concurrent development of theory and observation, related by appropriate methods of inference". An introductory economics textbook describes econometrics as allowing economists "to sift through mountains of data to extract simple relationships". Jan Tinbergen is one of the two founding fathers of econometrics. The other, Ragnar Frisch, also coined the term in the sense in which it is used today.
A basic tool for econometrics is the multiple linear regression model. Econometric theory uses statistical theory and mathematical statistics to evaluate and develop econometric methods. Econometricians try to find estimators that have desirable statistical properties including unbiasedness, efficiency, and consistency. Applied econometrics uses theoretical econometrics and real-world data for assessing economic theories, developing econometric models, analysing economic history, and forecasting.
Basic models: linear regression
A basic tool for econometrics is the multiple linear regression model. In modern econometrics, other statistical tools are frequently used, but linear regression is still the most frequently used starting point for an analysis. Estimating a linear regression on two variables can be visualised as fitting a line through data points representing paired values of the independent and dependent variables.
For example, consider Okun's law, which relates GDP growth to the unemployment rate. This relationship is represented in a linear regression where the change in unemployment rate () is a function of an intercept (), a given value of GDP growth multiplied by a slope coefficient and an error term, : | [
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10390_1 | The unknown parameters and can be estimated. Here is estimated to be 0.83 and is estimated to be -1.77. This means that if GDP growth increased by one percentage point, the unemployment rate would be predicted to drop by 1.77 * 1 points, other things held constant. The model could then be tested for statistical significance as to whether an increase in GDP growth is associated with a decrease in the unemployment, as hypothesized. If the estimate of were not significantly different from 0, the test would fail to find evidence that changes in the growth rate and unemployment rate were related. The variance in a prediction of the dependent variable (unemployment) as a function of the independent variable (GDP growth) is given in polynomial least squares.
Theory
Econometric theory uses statistical theory and mathematical statistics to evaluate and develop econometric methods. Econometricians try to find estimators that have desirable statistical properties including unbiasedness, efficiency, and consistency. An estimator is unbiased if its expected value is the true value of the parameter; it is consistent if it converges to the true value as the sample size gets larger, and it is efficient if the estimator has lower standard error than other unbiased estimators for a given sample size. Ordinary least squares (OLS) is often used for estimation since it provides the BLUE or "best linear unbiased estimator" (where "best" means most efficient, unbiased estimator) given the Gauss-Markov assumptions. When these assumptions are violated or other statistical properties are desired, other estimation techniques such as maximum likelihood estimation, generalized method of moments, or generalized least squares are used. Estimators that incorporate prior beliefs are advocated by those who favour Bayesian statistics over traditional, classical or "frequentist" approaches.
Methods
Applied econometrics uses theoretical econometrics and real-world data for assessing economic theories, developing econometric models, analysing economic history, and forecasting. | [
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10390_2 | Econometrics may use standard statistical models to study economic questions, but most often they are with observational data, rather than in controlled experiments. In this, the design of observational studies in econometrics is similar to the design of studies in other observational disciplines, such as astronomy, epidemiology, sociology and political science. Analysis of data from an observational study is guided by the study protocol, although exploratory data analysis may be useful for generating new hypotheses. Economics often analyses systems of equations and inequalities, such as supply and demand hypothesized to be in equilibrium. Consequently, the field of econometrics has developed methods for identification and estimation of simultaneous equations models. These methods are analogous to methods used in other areas of science, such as the field of system identification in systems analysis and control theory. Such methods may allow researchers to estimate models and investigate their empirical consequences, without directly manipulating the system.
One of the fundamental statistical methods used by econometricians is regression analysis. Regression methods are important in econometrics because economists typically cannot use controlled experiments. Econometricians often seek illuminating natural experiments in the absence of evidence from controlled experiments. Observational data may be subject to omitted-variable bias and a list of other problems that must be addressed using causal analysis of simultaneous-equation models.
In addition to natural experiments, quasi-experimental methods have been used increasingly commonly by econometricians since the 1980s, in order to credibly identify causal effects.
Example
A simple example of a relationship in econometrics from the field of labour economics is:
This example assumes that the natural logarithm of a person's wage is a linear function of the number of years of education that person has acquired. The parameter measures the increase in the natural log of the wage attributable to one more year of education. The term is a random variable representing all other factors that may have direct influence on wage. The econometric goal is to estimate the parameters, under specific assumptions about the random variable . For example, if is uncorrelated with years of education, then the equation can be estimated with ordinary least squares. | [
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10390_3 | If the researcher could randomly assign people to different levels of education, the data set thus generated would allow estimation of the effect of changes in years of education on wages. In reality, those experiments cannot be conducted. Instead, the econometrician observes the years of education of and the wages paid to people who differ along many dimensions. Given this kind of data, the estimated coefficient on Years of Education in the equation above reflects both the effect of education on wages and the effect of other variables on wages, if those other variables were correlated with education. For example, people born in certain places may have higher wages and higher levels of education. Unless the econometrician controls for place of birth in the above equation, the effect of birthplace on wages may be falsely attributed to the effect of education on wages.
The most obvious way to control for birthplace is to include a measure of the effect of birthplace in the equation above. Exclusion of birthplace, together with the assumption that is uncorrelated with education produces a misspecified model. Another technique is to include in the equation additional set of measured covariates which are not instrumental variables, yet render identifiable. An overview of econometric methods used to study this problem were provided by Card (1999).
Journals
The main journals that publish work in econometrics are Econometrica, the Journal of Econometrics, The Review of Economics and Statistics, Econometric Theory, the Journal of Applied Econometrics, Econometric Reviews, The Econometrics Journal, and the Journal of Business & Economic Statistics.
Limitations and criticisms
Like other forms of statistical analysis, badly specified econometric models may show a spurious relationship where two variables are correlated but causally unrelated. In a study of the use of econometrics in major economics journals, McCloskey concluded that some economists report p-values (following the Fisherian tradition of tests of significance of point null-hypotheses) and neglect concerns of type II errors; some economists fail to report estimates of the size of effects (apart from statistical significance) and to discuss their economic importance. She also argues that some economists also fail to use economic reasoning for model selection, especially for deciding which variables to include in a regression. | [
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] |
10390_4 | In some cases, economic variables cannot be experimentally manipulated as treatments randomly assigned to subjects. In such cases, economists rely on observational studies, often using data sets with many strongly associated covariates, resulting in enormous numbers of models with similar explanatory ability but different covariates and regression estimates. Regarding the plurality of models compatible with observational data-sets, Edward Leamer urged that "professionals ... properly withhold belief until an inference can be shown to be adequately insensitive to the choice of assumptions".
See also
Augmented Dickey–Fuller test
Choice modelling
Cowles Foundation
Econometric software
Financial econometrics
Financial modeling
Granger causality
Important publications in econometrics
Macroeconomic model
Methodological individualism
Predetermined variables
Single-equation methods (econometrics)
Spatial econometrics
Unit root
Further reading
Econometric Theory book on Wikibooks
Giovannini, Enrico Understanding Economic Statistics, OECD Publishing, 2008,
References
External links
Journal of Financial Econometrics
Econometric Society
The Econometrics Journal
Econometric Links
Teaching Econometrics (Index by the Economics Network (UK))
Applied Econometric Association
The Society for Financial Econometrics
The interview with Clive Granger – Nobel winner in 2003, about econometrics | [
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10391_0 | Ellen Gezina Maria van Langen (born February 9, 1966) is a Dutch former middle distance runner, who specialised in the 800 metres. A talented but injury-ridden athlete, she was the 1992 Olympic Champion for the women's 800 meters. She is now the director of the FBK games (the Hengelo meeting).
Career
Van Langen was born in Oldenzaal, Overijssel. Before she started running she played football. She only started to run seriously at 20. In 1989, she won her first of four national championships in the 800 meters and went on to win a silver medal behind Ana Quirot (Cuba) at the World Student Games (Universiade), running 1:59.82. In 1990, she finished fourth in the final of the European Championships in Split, with a Dutch national record of 1:57.57, behind the East German athletes Sigrun Wodars (gold), Christine Wachtel (silver) and Liliya Nurutdinova (bronze) from the Soviet Union.
During the year 1991, she was troubled by an Achilles tendon injury. At the Tokyo World Championships she was eliminated in the heats. The next year, leading up to the 1992 Olympic Games in Barcelona, she defeated Tokyo gold medallist Liliya Nurutdinova at the Fanny Blankers-Koen Games in Hengelo on 28 June in 1:56.66, the fastest time of the season prior to the Games. Just two weeks before the Olympics, she ran another fast time of 1:56.92 at Hechtel.
At the 1992 Olympic Games in Barcelona, Van Langen won the Olympic title in the 800 metres in a time of 1:55.54, a time which is still the Dutch national record. Fearing her final sprint, her competitors including favorites Nurutdinova, Quirot, Ella Kovacs (Romania) and Maria Mutola (Mozambique) set a rapid pace, running the first lap in a very fast time of 55.73, with Van Langen only in 6th position. Entering the final stretch Nurutdinova, who had led the final from the start, had a slight lead but Van Langen moved through on the inside in the last 50m to win a surprise victory, beating Nurutdinova (silver) and Quirot (bronze). | [
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10391_1 | The year 1992 remained her top year, winning 10 of her 11 races over 800m. After her Olympic triumph, she was plagued by various injuries. Her best result after 1992 was a 6th-place finish at the 1995 World Championships at Gothenburg. The ongoing injuries prevented her from defending her Olympic title in 1996. She retired from the sport in 1998.
Van Langen later explained the secret of her success. "I think what I could do well is I could die very well in a race and still continue,” she said. “That is very hard, because it hurts running the 800 meters. You have to overcome some boundaries in yourself to continue when it hurts like hell. I was good at it. If the Olympic race would have been run by each athlete individual and the fastest time was the winner I would not have won." She added: "I was also good in tactics, looking around me and taking the right decisions."
After her gold medal win in 1992 the Amsterdam unemployment benefits office terminated her dole on grounds that she might earn money from her victory in the future. She has a degree in economics from the University of Amsterdam, and currently works as an event manager for Global Sports Communication.
References
External links
1966 births
Living people
Athletes (track and field) at the 1992 Summer Olympics
Dutch female middle-distance runners
Medalists at the 1992 Summer Olympics
Olympic athletes of the Netherlands
Olympic gold medalists for the Netherlands
People from Oldenzaal
Olympic gold medalists in athletics (track and field)
Universiade medalists in athletics (track and field)
Universiade silver medalists for the Netherlands
Medalists at the 1989 Summer Universiade | [
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10392_0 | Emacs Lisp is a dialect of the Lisp programming language used as a scripting language by Emacs (a text editor family most commonly associated with GNU Emacs and XEmacs). It is used for implementing most of the editing functionality built into Emacs, the remainder being written in C, as is the Lisp interpreter. Emacs Lisp is also termed Elisp, although there is also an older, unrelated Lisp dialect with that name.
Users of Emacs commonly write Emacs Lisp code to customize and extend Emacs. Other options include the Customize feature that's been in GNU Emacs since version 20. Itself written in Emacs Lisp, Customize provides a set of preferences pages allowing the user to set options and preview their effect in the running Emacs session. When the user saves their changes, Customize simply writes the necessary Emacs Lisp code to the user's config file, which can be set to a special file that only Customize uses, to avoid the possibility of altering the user's own file.
Emacs Lisp can also function as a scripting language, much like the Unix Bourne shell or Perl, by calling Emacs in batch mode. In this way it may be called from the command line or via an executable file, and its editing functions, such as buffers and movement commands are available to the program just as in the normal mode. No user interface is presented when Emacs is started in batch mode; it simply executes the passed-in script and exits, displaying any output from the script. | [
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10392_1 | Compared to other Lisp dialects
Emacs Lisp is most closely related to Maclisp, with some later influence from Common Lisp. It supports imperative and functional programming methods. Richard Stallman chose Lisp as the extension language for his rewrite of Emacs (the original used Text Editor and Corrector (TECO) as its extension language) because of its powerful features, including the ability to treat functions as data. Although the Common Lisp standard had yet to be formulated, Scheme existed at the time Stallman was rewriting Gosling Emacs into GNU Emacs. He chose not to use it because of its comparatively poor performance on workstations (as opposed to the minicomputers that were Emacs' traditional home), and he wanted to develop a dialect which he thought would be more easily optimized.
The Lisp dialect used in Emacs differs substantially from the more modern Common Lisp and Scheme dialects used for applications programming. A prominent characteristic of Emacs Lisp is in its use of dynamic rather than lexical scope by default. That is, a function may reference local variables in the scope it is called from, but not in the scope where it was defined. Recently, there has been an ongoing effort to update code to use lexical scoping, for reasons outlined below.
Example
To understand the logic behind Emacs Lisp, it is important to remember that there is an emphasis on providing data structures and features specific to making a versatile text editor over implementing a general-purpose programming language. For example, Emacs Lisp cannot easily read a file a line at a time—the entire file must be read into an Emacs buffer. However, Emacs Lisp provides many features for navigating and modifying buffer text at a sentence, paragraph, or higher syntactic level as defined by modes.
Here follows a simple example of an Emacs extension written in Emacs Lisp. In Emacs, the editing area can be split into separate areas called windows, each displaying a different buffer. A buffer is a region of text loaded into Emacs' memory (possibly from a file) which can be saved into a text document. | [
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10392_2 | Users can press the default C-x 2 key binding to open a new window. This runs the Emacs Lisp function split-window-below. Normally, when the new window appears, it displays the same buffer as the previous one. Suppose we wish to make it display the next available buffer. In order to do this, the user writes the following Emacs Lisp code, in either an existing Emacs Lisp source file or an empty Emacs buffer:
(defun my-split-window-func ()
(interactive)
(split-window-below)
(set-window-buffer (next-window) (other-buffer)))
(global-set-key (kbd "C-x 2") #'my-split-window-func)
The first statement, (defun ...), defines a new function, my-split-window-func, which calls split-window-below (the old window-splitting function), then tells the new window to display another (new) buffer. The second statement, (global-set-key ...) re-binds the key sequence "C-x 2" to the new function.
This can also be written using the feature called advice, which allows the user to create wrappers around existing functions instead of defining their own. This has the advantage of not requiring keybindings to be changed and working wherever the original function is called, as well as being simpler to write but the disadvantage of making debugging more complicated. For this reason, advice is not allowed in the source code of GNU Emacs, but if a user wishes, the advice feature can be used in their code to reimplement the above code as follows:
(defadvice split-window-below
(after my-window-splitting-advice first () activate)
(set-window-buffer (next-window) (other-buffer)))
This instructs split-window-below to execute the user-supplied code whenever it is called, before executing the rest of the function. Advice can also be specified to execute after the original function, around it—literally wrapping the original, or to conditionally execute the original function based on the results of the advice. | [
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10392_3 | Emacs 24.4 replaces this defadvice mechanism with advice-add, which is claimed to be more flexible and simpler. The advice above could be reimplemented using the new system as:
(defun switch-to-next-window-in-split ()
(set-window-buffer (next-window) (other-buffer)))
(advice-add 'split-window-below :before #'switch-to-next-window-in-split)
These changes take effect as soon as the code is evaluated. It is not necessary to recompile, restart Emacs, or even rehash a configuration file. If the code is saved into an Emacs init file, then Emacs will load the extension the next time it starts. Otherwise, the changes must be reevaluated manually when Emacs is restarted.
Source code
Emacs Lisp code is stored in filesystems as plain text files, by convention with the filename suffix ".el". The user's init file is an exception, often appearing as ".emacs" despite being evaluated as any Emacs Lisp code. Recent versions of Emacs ("recent" in a 40-year-old program meaning roughly any version released since the mid-1990s) will also load ~/.emacs.el and ~/.emacs.d/init.el. Additionally, users may specify any file to load as a config file on the command line, or explicitly state that no config file is to be loaded. When the files are loaded, an interpreter component of the Emacs program reads and parses the functions and variables, storing them in memory. They are then available to other editing functions, and to user commands. Functions and variables can be freely modified and redefined without restarting the editor or reloading the config file. | [
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10392_4 | In order to save time and memory space, much of the functionality of Emacs loads only when required. Each set of optional features shipped with Emacs is implemented by a collection of Emacs code called a package or library. For example, there is a library for highlighting keywords in program source code, and a library for playing the game of Tetris. Each library is implemented using one or more Emacs Lisp source files. Libraries can define one or more major modes to activate and control their function.
Emacs developers write certain functions in C. These are primitives, also termed built-in functions or subrs. Although primitives can be called from Lisp code, they can only be modified by editing the C source files and recompiling. In GNU Emacs, primitives are not available as external libraries; they are part of the Emacs executable. In XEmacs, runtime loading of such primitives is possible, using the operating system's support for dynamic linking. Functions may be written as primitives because they need access to external data and libraries not otherwise available from Emacs Lisp, or because they are called often enough that the comparative speed of C versus Emacs Lisp makes a worthwhile difference.
However, because errors in C code can easily lead to segmentation violations or to more subtle bugs, which crash the editor, and because writing C code that interacts correctly with the Emacs Lisp garbage collector is error-prone, the number of functions implemented as primitives is kept to a necessary minimum.
Byte code
Byte-compiling can make Emacs Lisp code execute faster. Emacs contains a compiler which can translate Emacs Lisp source files into a special representation termed bytecode. Emacs Lisp bytecode files have the filename suffix ".elc". Compared to source files, bytecode files load faster, occupy less space on the disk, use less memory when loaded, and run faster. | [
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10392_5 | Bytecode still runs more slowly than primitives, but functions loaded as bytecode can be easily modified and re-loaded. In addition, bytecode files are platform-independent. The standard Emacs Lisp code distributed with Emacs is loaded as bytecode, although the matching source files are usually provided for the user's reference as well. User-supplied extensions are typically not byte-compiled, as they are neither as large nor as computationally intensive.
Language features
Notably, the "cl-lib" package implements a fairly large subset of Common Lisp. This package replaces an earlier "cl" package, which would overwrite existing Emacs Lisp function definitions with ones more similar to those found in Common Lisp. The "cl-lib" package, on the other hand, follows Emacs Lisp style guidelines more closely and prefixes each function and macro it defines with "cl-" (e.g., cl-defun, which doesn't conflict with the name of the built-in defun), avoiding the unexpected changes in behavior that could occur whenever the "cl" package was loaded.
Emacs Lisp (unlike some other Lisp implementations) does not do tail-call optimization. Without this, tail recursions can eventually lead to stack overflow.
The apel library aids in writing portable Emacs Lisp code, with the help of the polysylabi platform bridge.
Emacs Lisp is a Lisp-2 meaning that it has a function namespace which is separate from the namespace it uses for other variables.
From dynamic to lexical scoping
Like MacLisp, Emacs Lisp uses dynamic scope, offering static (or lexical) as an option starting from version 24. It can be activated by setting the file local variable lexical-binding. Before this option was added, one could use the lexical-let macro from the (now deprecated) "cl" package to provide effective lexical scope. | [
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10392_6 | In dynamic scoping, if a programmer declares a variable within the scope of a function, it is available to subroutines called from within that function. Originally, this was intended as an optimization; lexical scoping was still uncommon and of uncertain performance. "I asked RMS when he was implementing emacs lisp why it was dynamically scoped and his exact reply was that lexical scope was too inefficient." Dynamic scoping was also meant to provide greater flexibility for user customizations. However, dynamic scoping has several disadvantages. Firstly, it can easily lead to bugs in large programs, due to unintended interactions between variables in different functions. Secondly, accessing variables under dynamic scoping is generally slower than under lexical scoping.
References
External links
, GNU Project
Articles with example Lisp (programming language) code
Lisp
Free compilers and interpreters
Lisp programming language family
Scripting languages
Text editors
Programming languages created in 1985 | [
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10393_0 | Edward George Earle Lytton Bulwer-Lytton, 1st Baron Lytton, PC (25 May 180318 January 1873) was an English writer and politician. He served as a Whig member of Parliament from 1831 to 1841 and a Conservative from 1851 to 1866. He was Secretary of State for the Colonies from June 1858 to June 1859, choosing Richard Clement Moody as founder of British Columbia. He declined the Crown of Greece in 1862 after King Otto abdicated. He was created Baron Lytton of Knebworth in 1866. Bulwer-Lytton's works sold and paid him well. He coined famous phrases like "the great unwashed", "pursuit of the almighty dollar", "the pen is mightier than the sword", "dweller on the threshold", and the opening phrase "It was a dark and stormy night." The sardonic Bulwer-Lytton Fiction Contest, held annually since 1982, claims to seek the "opening sentence of the worst of all possible novels".
Life
Bulwer was born on 25 May 1803 to General William Earle Bulwer of Heydon Hall and Wood Dalling, Norfolk and Elizabeth Barbara Lytton, daughter of Richard Warburton Lytton of Knebworth House, Hertfordshire. He had two older brothers, William Earle Lytton Bulwer (1799–1877) and Henry (1801–1872), later Lord Dalling and Bulwer.
His father died and his mother moved to London when he was four years old. When he was 15, a tutor named Wallington, who tutored him at Ealing, encouraged him to publish an immature work: Ishmael and Other Poems. Around this time, Bulwer fell in love, but the woman's father induced her to marry another man. She died about the time that Bulwer went to Cambridge and he stated that her loss affected all his subsequent life.
In 1822 Bulwer-Lytton entered Trinity College, Cambridge, where he met John Auldjo, but soon moved to Trinity Hall. In 1825 he won the Chancellor's Gold Medal for English verse. In the following year he took his BA degree and printed for private circulation a small volume of poems, Weeds and Wild Flowers. He purchased an army commission in 1826, but sold it in 1829 without serving. | [
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10393_1 | In August 1827, he married Rosina Doyle Wheeler (1802–1882), a noted Irish beauty, but against the wishes of his mother, who withdrew his allowance, forcing him to work for a living. They had two children, Emily Elizabeth Bulwer-Lytton (1828–1848), and (Edward) Robert Lytton Bulwer-Lytton, 1st Earl of Lytton (1831–1891) who became Governor-General and Viceroy of British India (1876–1880). His writing and political work strained their marriage and his infidelity embittered Rosina. In 1833 they separated acrimoniously and in 1836 the separation became legal. Three years later, Rosina published Cheveley, or the Man of Honour (1839), a near-libellous fiction satirising her husband's alleged hypocrisy. In June 1858, when her husband was standing as parliamentary candidate for Hertfordshire, she denounced him at the hustings. He retaliated by threatening her publishers, withholding her allowance and denying her access to their children. Finally he had her committed to a mental asylum, but she was released a few weeks later after a public outcry. This she chronicled in a memoir, A Blighted Life (1880). She continued attacking her husband's character for several years.
The death of Bulwer's mother in 1843 meant his "exhaustion of toil and study had been completed by great anxiety and grief," and by "about the January of 1844, I was thoroughly shattered." In his mother's room at Knebworth House, which he inherited, he "had inscribed above the mantelpiece a request that future generations preserve the room as his beloved mother had used it." It remains hardly changed to this day. On 20 February 1844, in accordance with his mother's will, he changed his surname from Bulwer to Bulwer-Lytton and assumed the arms of Lytton by royal licence. His widowed mother had done the same in 1811. His brothers remained plain "Bulwer". | [
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10393_2 | By chance Bulwer-Lytton encountered a copy of "Captain Claridge's work on the "Water Cure", as practised by Priessnitz, at Graefenberg", and "making allowances for certain exaggerations therein", pondered the option of travelling to Graefenberg, but preferred to find something closer to home, with access to his own doctors in case of failure: "I who scarcely lived through a day without leech or potion!". After reading a pamphlet by Doctor James Wilson, who operated a hydropathic establishment with James Manby Gully at Malvern, he stayed there for "some nine or ten weeks", after which he "continued the system some seven weeks longer under Doctor Weiss, at Petersham", then again at "Doctor Schmidt's magnificent hydropathic establishment at Boppart" (at the former Marienberg Convent at Boppard), after developing a cold and fever upon his return home.
When Otto, King of Greece abdicated in 1862, Bulwer-Lytton was offered the Greek Crown, but declined.
The English Rosicrucian society, founded in 1867 by Robert Wentworth Little, claimed Bulwer-Lytton as their "Grand Patron", but he wrote to the society complaining that he was "extremely surprised" by their use of the title, as he had "never sanctioned such." Nevertheless, a number of esoteric groups have continued to claim Bulwer-Lytton as their own, chiefly because some of his writings – such as the 1842 book Zanoni – have included Rosicrucian and other esoteric notions. According to the Fulham Football Club, he once resided in the original Craven Cottage, today the site of their stadium. | [
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10393_3 | Bulwer-Lytton had long suffered from a disease of the ear, and for the last two or three years of his life lived in Torquay nursing his health. After an operation to cure deafness, an abscess formed in the ear and burst; he endured intense pain for a week and died at 2 am on 18 January 1873, just short of his 70th birthday. The cause of death was unclear but it was thought the infection had affected his brain and caused a fit. Rosina outlived him by nine years. Against his wishes, Bulwer-Lytton was honoured with a burial in Westminster Abbey. His unfinished history Athens: Its Rise and Fall was published posthumously.
Political career
Bulwer began his political career as a follower of Jeremy Bentham. In 1831 he was elected member for St Ives, Cornwall, after which he was returned for Lincoln in 1832, and sat in Parliament for that city for nine years. He spoke in favour of the Reform Bill and took the lead in securing the reduction, after vainly essaying the repeal, of the newspaper stamp duties. His influence was perhaps most keenly felt when, on the Whigs' dismissal from office in 1834, he issued a pamphlet entitled A Letter to a Late Cabinet Minister on the Crisis. Lord Melbourne, then Prime Minister, offered him a lordship of the Admiralty, which he declined as likely to interfere with his activity as an author.
Bulwer was created a baronet, of Knebworth House in the County of Hertford, in the Baronetage of the United Kingdom, in 1838. In 1841, he left Parliament and spent much of his time in travel. He did not return to politics until 1852, when having differed from Lord John Russell over the Corn Laws, he stood for Hertfordshire as a Conservative. Bulwer-Lytton held that seat until 1866, when he was raised to the peerage as Baron Lytton of Knebworth in the County of Hertford. In 1858 he entered Lord Derby's government as Secretary of State for the Colonies, thus serving alongside his old friend Disraeli. He was comparatively inactive in the House of Lords. | [
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10393_4 | "Just prior to his government's defeat in 1859 the Secretary of State for the Colonies, Sir Edward Bulwer Lytton, notified Sir George Ferguson Bowen of his appointment as Governor of the new colony to be known as 'Queen's Land'." The draft letter was ranked #4 in the 'Top 150: Documenting Queensland' exhibition when it toured to venues around Queensland from February 2009 to April 2010. The exhibition was part of Queensland State Archives' events and exhibition program which contributed to the state's Q150 celebrations, marking the 150th anniversary of the separation of Queensland from New South Wales.
British Columbia
When news of the Fraser Canyon Gold Rush reached London, Bulwer-Lytton, as Secretary of State for the Colonies, requested that the War Office recommend a field officer, "a man of good judgement possessing a knowledge of mankind", to lead a Corps of 150 (later increased to 172) Royal Engineers, who had been selected for their "superior discipline and intelligence". The War Office chose Richard Clement Moody, and Lord Lytton, who described Moody as his "distinguished friend", accepted the nomination in view of Moody's military record, his success as Governor of the Falkland Islands, and the distinguished record of his father, Colonel Thomas Moody, Knight at the Colonial Office. Moody was charged to establish British order and transform the newly established Colony of British Columbia (1858–66) into the British Empire's "bulwark in the farthest west" and "found a second England on the shores of the Pacific." Lytton desired to send to the colony "representatives of the best of British culture, not just a police force": he sought men who possessed "courtesy, high breeding and urbane knowledge of the world," and decided to send Moody, whom the Government considered to be the archetypal "English gentleman and British Officer" at the head of the Royal Engineers, Columbia Detachment, to whom he wrote an impassioned letter.
The former HBC Fort Dallas at Camchin, the confluence of the Thompson and Fraser Rivers, was renamed in his honour by Governor Sir James Douglas in 1858 as Lytton, British Columbia. | [
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